# Forward utility and market adjustments in relative investment-consumption
games of many players ††thanks: The authors thank T. Zariphopoulou (U. of
Texas, US), M. Anthropelos (U. of Piraeus, GR), M. Mrad (U. Paris 13, FR) and
D. Lacker (Columbia U., US) for the helpful discussions.

Gonçalo dos Reisa,b,***G.d.R. acknowledges support from the _Fundaç
$\tilde{\text{a}}$o para a Ci$\hat{e}$ncia e a Tecnologia_ (Portuguese
Foundation for Science and Technology) through the project UIDB/00297/2020
(Centro de Matemática e Aplicaç$\tilde{\text{o}}$es CMA/FCT/UNL).
<EMAIL_ADDRESS>Vadim Platonova
<EMAIL_ADDRESS>
(a School of Mathematics, University of Edinburgh, The King’s Buildings,
Edinburgh, UK
b Centro de Matemática e Aplicaç$\tilde{\text{o}}$es (CMA), FCT, UNL, Portugal
() )

###### Abstract

We study a portfolio management problem featuring many-player and mean field
competition, investment and consumption, and relative performance concerns
under the forward performance processes (FPP) framework. We focus on agents
using power (CRRA) type FPPs for their investment-consumption optimisation
problem an under common noise Merton market model and we solve both the many-
player and mean field game providing closed-form expressions for the solutions
where the limit of the former yields the latter.

In our case, the FPP framework yields a continuum of solutions for the
consumption component as indexed to a market parameter we coin “market
consumption intensity”. The parameter permits the agent to set a preference
for their consumption going forward in time that, in the competition case,
reflects a common market behaviour. We show the FPP framework, under both
competition and no-competition, allows the agent to disentangle his risk-
tolerance and elasticity of intertemporal substitution (EIS) just like
Epstein-Zin preferences under recursive utility framework and unlike the
classical utility theory one. This, in turn, allows a finer analysis on the
agent’s consumption “income" and “substitution" regimes, and, of independent
interest, motivates a new strand of economics research on EIS under the FPP
framework.

We find that competition rescales the agent’s perception of consumption in a
non-trivial manner in addition to a time-dependent elasticity of conformity of
the agent to the market’s consumption intensity.

Keywords: optimal investment and consumption, forward performance criteria,
mean field games, relative performance, common noise Merton problem,
elasticity of intertemporal substitution

2020 AMS subject classifications:
Primary: 91G10, 91A30, 49N80. Secondary: 60G60, 91A06, 35Q93

JEL subject classifications:
G11, C73, P46, C68

Note to the reader: This version is the extended arXiv version of this work
containing all proofs and details. The final published version omits some of
the details.

##  1 Introduction

We study many-player games of wealth-consumption optimisation under the
forward performance framework in a Merton market model featuring common-noise.
In each game, the agents trade in a specific stock affected by a common market
signal and seek to optimise their wealth and consumption process while having
relative concerns towards the average wealth and consumption of the other
agents. The core features of our model are (a) relative consumption concerns,
(b) relative wealth concerns, (c) asset specialisation, (d) game competition
(finite-player and mean field games) and (e) the forward performance processes
(FPP) view. The former (a)–(d) have been analysed in [46] through the lens of
the classical utility framework. At the same time, here we subscribe to the
elegantly argued (supported by empirical evidence) FPP paradigm of [3] to view
the upstream [46] through its light. The arguments here using FPPs,
competition and the presence of consumption make the analysis involved
revealing elements not present in [46] (despite the similarity to (a)–(d)) or
[44, 60].

Throughout we consider fund managers that trade between their individual stock
and common riskless asset, so called problem of _asset specialisation_. This
theory developed from the initial works of [11], [18], [51] to further
developments in [64], [63], [41], [47, 46], [3]. These works report on agents’
asset specialisation for a variety of reasons, like familiarity, trading costs
and constraints, reduction of learning costs, industry specialisation.
Competition games, of $2$-player, $n$-player or mean field type, in the
context of asset specialisation, has received much attention recently and, to
avoid repeating the arguments there, we defer to the introductions of [47, 46,
22, 3] that relate specifically to our context. More generally to the
monographs [14] for a long-view of mean field games theory and applications.

The literature on portfolio management for agents with utility preferences
with performance concerns is a growing one, benchmarking is a feature of human
nature and critical for fund managers who need to keep the fund competitive.
We refer to the excellent economic and finance motivations found in [3] and
also [10, 47, 46]. In this work, we build on the structure proposed firstly in
[30] and then in [31], and afterwards expanded by [10, 47, 20, 46].
Additionally, we point the reader to the beautiful introductions of [47, 46]
who brought those concepts to the framework of mean field games. Further,
those works also make for an excellent review of mean field games in the
context of the Merton problem, which is the framework underlying our work.
Investment under performance concerns has been taken up in many variants, most
recently: [20] analyse a partial information finite-player CARA utility game
with performance concerns employing Forward Backward SDE (FBSDE) machinery
(market model with stochastic coefficients) and applying deep learning
techniques for their numerical study, see [20, Table 1]. [32] work in the same
context but address the mean field game in the full non-Markovian framework
and using mean field FBSDEs, see also [32, Table 1]. Aspects of investment
under relative consumption are much less explored, only in [46].

At the core of our work, are the ideas behind the forward investment
performance criteria introduced and developed by Musiela and Zariphopoulou
[57, 54] and [39] as a way to solve portfolio optimisation problems without
two specific drawbacks from the classical utility theory. When entering the
market, investors prescribe their risk profile at horizon time and therefore
cannot adapt it to changes in market conditions or update risk preferences;
additionally, the investment time horizon is fixed, and the portfolio is
derived in respect to this temporal reference point. The forward criteria
(portfolio) optimisation problem is formulated as a maximisation of the
conditional expectation of a stochastic utility function across all points in
time. Hence, time-horizons are arbitrary, which is more realistic for
practitioners. In essence, the forward dynamic performance map is built with
the property of dynamic consistency taken as the starting point and then
deriving a certain consistency condition.

Forward performance processes (FPP), as defined in Musiela and Zariphopoulou
[52], capture the time evolution of such stochastic utility functions and
since then much progress has been done in characterising such processes in a
variety of settings and assumptions [9], [25], [66], [55], [2], [26], [27],
[16], [35], [43], [42], [49]. For excellent literature overviews on the
developments of FPP we point the reader to [15], [4], [50] and [42].
Particularly clear is [42] who provides a near taxonomic breakdown of the
literature of the forward criteria in respect to their behaviour in time and
associated explored approaches to then discuss an optimal investment and
consumption problem. Here, as in [42] or [4], we focus on a class of forward
investment-consumption criteria with the specific property that they are
differentiable in time (a dynamics without volatility). In [4] the authors
study an optimal portfolio selection problem (no consumption, no competition)
under the FPP of power form in an incomplete market. Their arguments make use
of HJB machinery and, like in this work or [16], an assumption of "separable
power factor form" for the FPP is made (Assumption 4.3 below). Lastly, a
general form of the utility dynamics is made with a non-zero diffusion
component and the driving market model is fully stochastic multi-factor one
coined "EVE correlation" models. The study of consumption with FPPs (or
forward utility) was considered in [9, 26, 16]; interesting is [16] who manage
the difficulty associated to having a FPP dynamics featuring a volatility
component by assuming the volatility to be exogenously postulated by the agent
as a market perception; their FPP consumption optimisation problem yields non-
uniqueness.

Competition within the framework of FPP is in its infancy. The concept
originated in [34] for a two-player game in the CRRA context with a Merton
market model, and its vision is expanded upon in [3], still within a two-
player game but allowing for random coefficients in an incomplete market
model. A formulation of mean field forward performance games and the concept
of mean field forward equilibrium were proposed in [22], by merging [34] and
[47], where also the finite-player game is addressed within the CARA utility
framework. In these three works, the forward performance dynamics follows an
ODE (i.e., there is no volatility). From a competition perspective, the FPP
approach reflects that agents need not all have the same time-horizons since,
in particular, under asset specialisation different industries have different
timeframes (although quarterly reporting are common points of reference), see
[3] for a full discussion.

The FPP framework is not the only theory seeking to overcome limitations of
the classical utility theory. An alternative from the 90s are so-called
Epstein-Zin preferences within the recursive utility framework [28], [29],
[65] expanding on the theoretical framework of Kreps-Porteus [45]. For the
consumption problem, the classic CRRA utility optimisation yields a strict
relation between the agent’s risk tolerance $\delta$ and Elasticity of
intertemporal substitution (EIS), namely that $\textrm{EIS}^{\textrm{classic,
\cite[cite]{[\@@bibref{Number}{lackersoret2020many}{}{}]}}}=\delta$ (e.g., see
[46]) and disentangling both was at heart the Epstein-Zin preferences. As for
EIS and as argued in [62], “higher interest rates increase the overall wealth
of the consumer due to higher cash-flows in future periods. The effect that
consumers spend a part of this higher future income already today is called
“income effect”. On the other hand, with higher interest rates a smaller
fraction of today’s consumption has to be saved in order to have an additional
unit of consumption tomorrow. This motive to save more today and postpone
today’s consumption is called the “substitution effect”. Consumers with a high
EIS are more willing to substitute consumption over time, which has a direct
impact on the substitution effect”. Now, classic theory yields
$\textrm{EIS}^{\textrm{classic,
\cite[cite]{[\@@bibref{Number}{lackersoret2020many}{}{}]}}}=\delta$, however,
the risk tolerance is atemporal relating how a consumer substitutes
consumption across different states of the world while EIS is intertemporal
relating how a consumer substitutes consumption between now and later [19].
Thus classic utility framework cannot capture how agent competition, with or
without performance concerns, changes EIS.

Full time-continuous stochastic preferences capturing “intertemporal
substitution” of Hindy-Huang-Kreps [40] have been used to understand
consumption optimisation [8, 7] and Arrow-Debreu equilibria [6] using Lévy
processes and singular stochastic control to discuss. More recently, [1]
revisited the EIS discussion using [23, 24]’s stochastic recursive utility in
continuous time specified to the Kreps-Porteus [45] family. Forthcoming is
[36] (also [37]) who recast the MFG of [47] under the Epstein-Zin / Kreps-
Porteus recursive utility framework. Nonetheless, inspecting these reference
one sees a few of the criticism aimed at the standard utility framework, as
highlighted in [3], appearing again. E.g., in [12] we see a dependence on
investment’s horizon “The investor’s horizon also plays a crucial role in
optimal policies” and that the underlying model is fixed the investment
timeframe.

With this work, we investigate an $n$-player and mean field game for asset
specialising agents who optimise wealth-consumption with relative wealth-
consumption concerns through the power of the FPP framework. The tractability
of our setting with FPPs yields findings not seen in the classic utility
theory. Namely, our contributions are

(I) We expand the concept of CRRA FPP to a game-theoretical framework both for
finite-player games and mean field games working for tractability within the
common-noise Merton market model. The MFG approach is based on the simpler
concept work [22] for CARA FPP (wealth optimisation only), and [46] for the
classical CRRA investment-consumption utility problem. For both games, we
provide explicit expressions for the quantities of interest. From a
methodological point of view, we solve the control problem by combining convex
duality with HJB-type arguments.

(II) With the inclusion of consumption, the CRRA FPP wealth-consumption
problem does not have a unique solution but a class of them indexed by some
$\kappa\in\mathbb{R}$ interpreted as _market consumption intensity_. This
parameter is present in the single-agent optimisation (no
competition/performance concerns) and both games settings. It is common to all
players and encapsulates the time-weight the environment assigns to the
utility from consumption when compared to the accumulation of wealth. The
tractability of our setting, exposes the $\kappa$ parameter to clearly enable
an extra layer of interpretable market modelling (see Section 5) and,
critically, when $\kappa=0$ we recover classic utility theory results [46]
also [44, 60].

(III) We show that the CRRA FPP framework (even without competition)
encapsulates the same key feature of the Epstein-Zin preferences recursive
utility. It breaks the strict relation between risk tolerance $\delta$ and
Elasticity of intertemporal substitution (EIS) of the classical utility
theory, and at the same time keeps its core features of independence from the
investment time-horizon and flexibility towards updating risk preferences.
These features stand to span a fresh breath of new economics studies on EIS in
the scope of FPP (and its features).

We show that competition/performance concerns in our model ($\theta\neq
0,\leavevmode\nobreak\ \theta\in(0,1]$) additionally re-scales the agent’s
perception of consumption, namely,

$\displaystyle\textrm{EIS}^{\textrm{with competition}(\theta\neq
0)}_{t}=\frac{\textrm{EIS}^{\textrm{no
competition}(\theta=0)}}{1+\kappa\theta(1-\delta)\frac{1}{\gamma_{t}}},\quad\textrm{where}\quad\textrm{EIS}^{\textrm{no
competition}(\theta=0)}=(1-\kappa)\delta,$

compared with $\textrm{EIS}^{\textrm{classic,
\cite[cite]{[\@@bibref{Number}{lackersoret2020many}{}{}]}}}=\delta$, where
$(\gamma_{t})_{t\geq 0}$ represents the _elasticity of conformity_ of the
agent in relation to the market and is time dependent, $\delta>0$ is the
agent’s risk tolerance and $\kappa\in\mathbb{R}$ is market consumption
intensity that affects all agents in the market. [37] finds related results
using Epstein-Zin recursive utility.

Organisation of the paper. In Section 2 we introduce the financial market and
notation. In Sections 3 and 4 we study the finite-agent and mean field game
respectively, and in each part we study particular tractable case allowing for
explicit calculations and examples. The interpretations of equilibria,
parameter relations and the models are left for the final Section 5.

##  2 Asset specialisation and relative performance concerns

We formulate our working market models mimicking the construction presented in
[47, 46, 22].

The market. The market environment has one riskless asset and $n$ risky
securities which serve as proxies for two distinct asset classes. We assume
their prices to be of log-normal type, each driven by two independent Brownian
motions. More precisely the price $(S^{i}_{t})_{t\geqslant 0}$ of the stock
$i$ traded exclusively by the $i$-th agent solves

$\displaystyle\frac{dS_{t}^{i}}{S_{t}^{i}}$
$\displaystyle=\mu_{i}dt+\nu_{i}dW_{t}^{i}+\sigma_{i}dB_{t},$ (2.1)

with constant parameters $\mu_{i}>0$, $\sigma_{i}\geqslant 0$ and
$\nu_{i}\geqslant 0$ with $\sigma_{i}+\nu_{i}>0$. We refer the reader to [47,
46] for an in-depth motivation of the model. The one-dimensional standard
Brownian motions $B,W^{1},\dots,W^{n}$ are independent. When $\sigma_{i}>0$,
the process $B$ induces a correlation between the stocks, and thus we call $B$
the _common noise_ and $W^{i}$ an _idiosyncratic noise_. The independent
Brownian motions $B,W^{1},\dots,W^{n}$ are defined on a probability space
$(\Omega,\mathbb{F},\mathcal{F},\mathbb{P})$ endowed with the natural
filtration $\mathbb{F}=(\mathcal{F}_{t})_{t\geqslant 0}$ generated by them and
satisfies the usual conditions.

We recall the case of single common stock, where for any
$i=1,\dots,n,\leavevmode\nobreak\
(\mu_{i},\sigma_{i})=(\mu,\sigma),\leavevmode\nobreak\ \nu_{i}=0,$ for some
$\mu,\sigma>0$ and independent of $i$. The single common stock case has been
explored in great generality in [30, 31, 10] incorporating portfolio
constraints, general stock price dynamics and risk-sharing mechanisms.

We aim to contribute to the literature on mean field games and FPP by
providing an explicitly solvable example. As argued by [46], outside linear-
quadratic structures such is very rare, and it is one of these rarities we
bring here. We work with the very tractable model (2.1) and include common
noise, heterogeneous agents, a mean field interaction through the controls in
addition to the state processes and FPPs.

Agents’ wealth. Each agent $i=1,\ldots,n$ trades using a self-financing
strategy, $(\pi_{t}^{i})_{t\geqslant 0}$, representing the (discounted by the
bond) fraction of wealth invested in the $i$-th stock and consumption policy
$(c^{i}_{t})_{t\geqslant 0}$, representing the instantaneous rate of
consumption per unit of wealth. The $i^{th}$ agent’s wealth dynamics
$(X^{i}_{t})_{t\geqslant 0}$ is given by

$dX_{t}^{i}=\pi_{t}^{i}X_{t}^{i}\Big{(}\mu_{i}dt+\nu_{i}dW_{t}^{i}+\sigma_{i}dB_{t}\Big{)}-c^{i}_{t}X^{i}_{t}dt,\qquad\textrm{with}\quad
X_{0}^{i}=x_{0}^{i}\in\mathbb{R}.$ (2.2)

A portfolio investment-consumption strategy is deemed admissible if it belongs
to the admissibility set $\mathcal{A}^{i}$, defined as

$\displaystyle\mathcal{A}^{i}=\Big{\\{}(\pi^{i},c^{i})$
$\displaystyle:\textrm{$\mathbb{F}$-progressively measurable
$\mathbb{R}\times\mathbb{R}_{+}$-valued process
}(\pi^{i}_{t},c^{i}_{t})_{t\geqslant 0},$ $\displaystyle\qquad\textrm{such
that
}\mathbb{E}\Big{[}\int_{0}^{t}(|\pi_{s}|^{2}+|c_{s}|^{2})ds\Big{]}<\infty,\
\textrm{for any }t>0\Big{\\}}.$

The Agents’ interaction and relative performance concerns. Each manager
measures the performance of her strategy taking into account the policies of
the others. Each agent engages in a form of social interaction that affects
that agent’s perception of wealth, all in a multiplicative fashion modelled
through the geometric average wealth of all agents. The _relative performance
metric_ of manager $i\in\\{1,\dots,n\\}$, denoted $\breve{X}^{i}$ is defined
to be

$\displaystyle\breve{X}^{i}=\frac{X^{i}}{{\widetilde{X}}^{\theta_{i}}},\quad\textrm{where}\quad\widetilde{X}:=\Big{(}\prod_{k=1}^{n}X^{k}\Big{)}^{\frac{1}{n}}\quad\textrm{and}\quad\theta_{i}\in[0,1].$
(2.3)

Similarly to [47, Remark 3.3], [46, Section 2] or [22, Section 2], it is
natural to replace the average wealth $\widetilde{X}$ in (2.3) by the average
over all other agents. With that in mind we define for convenience
$\widetilde{X}^{(-i)}=\Big{(}\prod_{k\neq i}X^{k}\Big{)}^{\frac{1}{n-1}}$.
This leads us further to a second version of (2.3), the relative performance
wealth process,

$\displaystyle\widehat{X}^{i}=\frac{X^{i}}{{\Big{(}\widetilde{X}^{(-i)}}\Big{)}^{\theta_{i}}},\qquad\textrm{where}\qquad\widetilde{X}^{(-i)}=\Big{(}\prod_{k\neq
i}^{n}X^{k}\Big{)}^{\frac{1}{n-1}}.$ (2.4)

In the same vein we introduce the _relative consumption metric_

$\displaystyle\hat{c}^{i}=\dfrac{c^{i}}{\Big{(}\tilde{c}^{(-i)}\Big{)}^{\theta_{i}}},\quad\text{where}\quad\tilde{c}^{(-i)}=\Big{(}\prod_{k\neq
i}^{n}c^{k}\Big{)}^{\frac{1}{n-1}}.$ (2.5)

We obtain the dynamics for _the average performance wealth_
$Y=\widetilde{X}^{(-i)}$,

$\displaystyle\frac{dY_{t}}{Y_{t}}=(\eta_{t}-\bar{c}_{t}^{(-i)})dt+\frac{1}{n-1}\sum_{k\neq
i}^{n}\nu_{k}\pi_{k}dW_{t}^{k}+\overline{\sigma\pi}^{(-i)}dB_{t},\quad
Y_{0}=\bigg{(}\prod_{k\neq i}^{n}x_{0}^{k}\bigg{)}^{\frac{1}{n-1}},$

where we define the helpful auxiliary quantities

$\displaystyle\eta:=\overline{\mu\pi}^{(-i)}-\frac{1}{2}\left(\overline{\Sigma\pi^{2}}^{(-i)}-\big{(}\overline{\sigma\pi}^{(-i)}\big{)}^{2}-\frac{1}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\right),\quad\overline{\mu\pi}^{(-i)}:=\frac{1}{n-1}\sum_{k\neq
i}^{n}\mu_{k}\pi^{k}_{t},$
$\displaystyle\overline{(\nu\pi)^{2}}^{(-i)}=\frac{1}{n-1}\sum_{k\neq
i}^{n}(\nu_{k}\pi^{k}_{t})^{2},\quad\overline{\sigma\pi}^{(-i)}:=\frac{1}{n-1}\sum_{k\neq
i}^{n}\sigma_{k}\pi^{k}_{t},\quad\overline{\Sigma\pi^{2}}^{(-i)}:=\frac{1}{n-1}\sum_{k\neq
i}^{n}\Sigma_{k}(\pi^{k}_{t})^{2},$
$\displaystyle\qquad\bar{c}^{(-i)}_{t}=\frac{1}{n-1}\sum_{k\neq
i}^{n}c^{k}_{t},\quad\Sigma_{k}=\sigma_{k}^{2}+\nu_{k}^{2}.$

Via Itô’s formula one finds the dynamics of _relative performance wealth_
$\widehat{X}^{i}$ to be

$\displaystyle\frac{d\widehat{X}^{i}_{t}}{\widehat{X}^{i}_{t}}=\xi_{i}dt-\big{(}c^{i}_{t}-\theta_{i}\bar{c}_{t}^{(-i)}\big{)}+\bigg{(}\nu_{i}\pi^{i}_{t}dW^{i}_{t}-\theta_{i}\Big{(}\frac{1}{n-1}\sum_{k\neq
i}^{n}\nu_{k}\pi^{k}_{t}dW^{k}_{t}\Big{)}\bigg{)}+\bigg{(}\sigma_{i}\pi^{i}_{t}-\theta_{i}\overline{\sigma\pi}^{(-i)}\bigg{)}dB_{t},$
(2.6)

where

$\displaystyle\xi_{i}:=\mu_{i}\pi^{i}_{t}-\theta_{i}\overline{\mu\pi}^{(-i)}+\frac{\theta_{i}}{2}\overline{\Sigma\pi^{2}}^{(-i)}+\frac{\theta_{i}^{2}}{2}\bigg{(}\big{(}\overline{\sigma\pi}^{(-i)}\big{)}^{2}+\frac{1}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\bigg{)}-\theta_{i}\sigma_{i}\pi^{i}_{t}\overline{\sigma\pi}^{(-i)}.$

##  3 The finite player forward optimisation game

###  3.1 Forward relative performance

Each manager $i\in\\{1,\cdots,n\\}$ measures the output of her relative
performance metric using a forward relative utility as modelled by an
$\mathcal{F}_{t}$-progressively measurable random field
$Q^{i}:\Omega\times\mathbb{R}\times[0,\infty)\to\mathbb{R}$ for
$i\in\\{1,\dots,n\\}$. The below criteria follows those proposed in [34]
(power-type) and later in [22] (exponential-type) for a forward-performance
game.

The formulation here is inspired in the first step of the usual strategy of
solving a Nash equilibria games, namely, the best response of an agent to the
actions of all other agents. Take manager $i$ and assume all other agents
$j\neq i$ have acted with a policy $(\pi^{j},c^{j})$ then for any strategy
$(\pi^{i},c^{i})\in\mathcal{A}^{i}$, the process
$Q^{i}(\widehat{X}^{i}_{t},t)$ is a (local) supermartingale, and there exists
$\pi^{i,*},c^{i,*}\in\mathcal{A}^{i}$ such that
$Q^{i}(\widehat{X}^{i,*}_{t},t)$ is a (local) martingale where
$\widehat{X}^{i}$ and $\widehat{X}^{i,*}$ solves (2.4) with strategies
$(\pi^{i},c^{i})$ and $(\pi^{i,*},c^{i,*})$ respectively.

This version of a relative criterion is (implicit and) endogenously
parametrised by the policies of all other managers $j\neq i$ over which there
is no assumption on their optimality. In Nash-game language, we solve the so-
called best response.

######  Definition 3.1 (Forward relative performance for the manager).

Each manager $i\in\\{1,\dots,n\\}$ satisfies the following conditions: let for
any $j\neq i,\leavevmode\nobreak\ (\pi^{j},c^{j})\in\mathcal{A}^{j}$ be
arbitrary but fixed and admissible policies, in other words, the other
managers have fixed their investment-consumption admissible strategies.

For $(\pi^{i},c^{i})\in\mathcal{A}^{i}$, define the $\mathbb{F}$-progressively
measurable random field
$Q^{i}:\Omega\times\mathbb{R}\times[0,\infty)\to\mathbb{R}$ as

$\displaystyle
Q^{i}(x,t):=U^{i}(x,t)+\int_{0}^{t}V^{i}(\hat{c}_{s}^{i}x,s)ds,$

where $\hat{c}^{i}$ is given by (2.5) and
$U^{i},V^{i}:\Omega\times\mathbb{R}\times[0,\infty)\to\mathbb{R}$ are two
other $\mathbb{F}$-progressively measurable random fields.

The random field $Q^{i}$ is a _forward relative performance process for $i$-th
manager_ if, for all $t\geqslant 0$, the following conditions hold:

  1. i)

The mappings $x\mapsto U^{i}(x,t)$ and $y\mapsto V^{i}(y,t)$ are
$\mathbb{P}$-a.s. strictly increasing and strictly concave;

  2. ii)

For any $(\pi^{i,*},c^{i,*})\in\mathcal{A}^{i},\leavevmode\nobreak\
Q^{i}(\widehat{X}^{i}_{t},t)$ is a (local) supermartingale, where
$\widehat{X}^{i}$ is the relative performance wealth process given in (2.4);

  3. iii)

There exists $(\pi^{i,*},c^{i,*})\in\mathcal{A}^{i}$ such that
$Q^{i}(\widehat{X}^{i,*}_{t},t)$ is a (local) martingale, where
$\widehat{X}^{i,*}$ solves (2.4) with the strategies $(\pi^{i,*},c^{i,*})$
being used. The strategy $(\pi^{i,*},c^{i,*})$ is said to be optimal.

In the above definition, we do not make explicit references to the initial
conditions $U^{i}(x,0)$, $V^{i}(x,0)$ but we assume that admissible
($\mathcal{F}_{0}$-measurable) initial data exists such that the above
definition is viable. Contrary to the classical expected utility case, the
forward performance process is an manager-specific input. Once it is chosen,
the supermartingale and martingale properties impose conditions on the drift
of the process. Under enough regularity, these conditions lead to the forward
performance SPDE (see [56, 25]) which, in our case, reduces to a PDE with
stochastic coefficients (see Proposition 3.4 below).

Since we are working in a log-normal market, it suffices to study smooth
relative performance criteria of zero volatility (of the FPP map). Such
processes are extensively analysed in [53] in the absence of relative
performance concerns. There, a concise characterisation of the forward
criteria is given along necessary and sufficient conditions for their
existence and uniqueness. In that setting, the zero-volatility forward
processes are always time-decreasing processes. We point to the reader that
this does not have to be case if relative performance concerns are present
(see also [34, 22]). Before proving the main result of the subsection, we make
a standing assumption regarding the regularity of the FPP map.

######  Assumption 3.2.

Assume that the partial derivatives $U^{i}_{t}(x,t),\leavevmode\nobreak\
U^{i}_{x}(x,t),\leavevmode\nobreak\ U^{i}_{xx}(x,t)$ and
$V^{i}_{y}(y,t),V^{i}_{yy}(y,t)$ exist for all $t\geqslant
0,\leavevmode\nobreak\ x,y\in\mathbb{R},\leavevmode\nobreak\ \mathbb{P}$-a.s.

Furthermore, $x\mapsto U(x,t)$ and $y\mapsto V(y,t)$ are strictly increasing
($U_{x},V_{y}>0$) and strictly concave ($U_{xx},V_{yy}<0$) for any $t\geqslant
0,\leavevmode\nobreak\ x,y\in\mathbb{R},\leavevmode\nobreak\ \mathbb{P}$-a.s.

From Assumption 3.2, the Itô decomposition of the forward map is

$\displaystyle
dQ^{i}(x,t)=U^{i}_{t}(x,t)dt+V^{i}(\hat{c}_{t}^{i}x,t)dt,\quad\textrm{for
}i\in\\{1,\dots,n\\},\qquad Q^{i}(x,0)=u^{i}_{0}(x).$ (3.1)

For posterior use, we recall the notion of Fenchel-Legendre transform applied
to a random field $V$ under the above assumption.

######  Definition 3.3.

Let $V:\Omega\times\mathbb{R}\times[0,\infty)\to\mathbb{R}$ be a random field
such that $y\mapsto V(y,t)$ is a $\mathbb{P}$-a.s. strictly concave function
for all $t\geq 0$. Define the random field
$\widetilde{V}:\Omega\times\mathbb{R}\times[0,\infty)\to\mathbb{R}$ as
$\widetilde{V}(y,t):=\sup_{x\in\mathbb{R}}\big{\\{}V(x,t)-yx\big{\\}}$ for any
$y\in\mathbb{R}$, $t\geq 0$ and $\omega\in\Omega$.

Then, we call $\widetilde{V}$ the _Fenchel-Legendre transform_ of $V$.

For a map $V$ satisfying Assumption 3.2, its differentiability and strict
concavity ensures $\widetilde{V}$ is always well-defined and can be computed
(up to a closed form).

###  3.2 Best responses

Here we derive a PDE with random coefficients and an optimal investment-
consumption strategy for a smooth relative performance criteria of zero-
volatility of some agent $i$ assuming that all other agents $j\neq i$ have
made their investment decisions.

######  Proposition 3.4 (Best responses).

Fix $i\in\\{1,\dots,n\\}$. Assume that each manager $j\neq i$ follows
$(\pi^{j},c^{j})\in\mathcal{A}^{j}$. Consider the PDE with stochastic
coefficients for $(x,t)\in\mathbb{R}\times[0,\infty)$ given by

$\displaystyle U^{i}_{t}$
$\displaystyle=\bigg{(}\theta_{i}\overline{\mu\pi}^{(-i)}-\frac{\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}(\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)})}{\nu_{i}^{2}+\sigma_{i}^{2}}-\frac{\theta_{i}}{2}\overline{\Sigma\pi^{2}}^{(-i)}-\frac{\theta_{i}^{2}}{2}\big{(}\big{(}\overline{\sigma\pi}^{(-i)}\big{)}^{2}+\frac{1}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\big{)}\bigg{)}xU^{i}_{x}$
$\displaystyle\qquad\qquad+\frac{(\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)})^{2}}{2(\nu_{i}^{2}+\sigma_{i}^{2})}\frac{(U^{i}_{x})^{2}}{U^{i}_{xx}}+\frac{1}{2}\bigg{(}\Big{(}\theta_{i}\overline{\sigma\pi}^{(-i)}\Big{)}^{2}\Big{(}\frac{\sigma_{i}^{2}}{\nu_{i}^{2}+\sigma_{i}^{2}}-1\Big{)}-\frac{\theta_{i}^{2}}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\bigg{)}x^{2}U^{i}_{xx}$
$\displaystyle\qquad\qquad+\theta_{i}\bar{c}_{t}^{(-i)}U_{x}^{i}-\widetilde{V}^{i}(U^{i}_{x},t),$
(3.2)

where $\widetilde{V}^{i}$ stands for Fenchel-Legendre transform of $V^{i}$ in
variable $y$. Assume that for admissible initial conditions
$U^{i}(\cdot,0)=u^{i}_{0}(\cdot),\leavevmode\nobreak\
V^{i}(\cdot,0)=v^{i}_{0}(\cdot)$, the PDE has a smooth solution
$(U^{i},V^{i})$, that is not necessarily unique, but satisfy Assumption 3.2.

Define the strategy $(\pi^{i,*},c^{i,*})$

$\displaystyle\pi^{i,*}_{t}$
$\displaystyle=\frac{1}{\nu_{i}^{2}+\sigma_{i}^{2}}\Big{(}{\theta_{i}\sigma_{i}\overline{(\sigma\pi)}^{(-i)}_{t}}-\big{(}\mu_{i}-\theta_{i}\sigma_{i}\overline{(\sigma\pi)}^{(-i)}_{t}\big{)}\frac{U^{i}_{x}(\widehat{X}^{i,*}_{t},t)}{U^{i}_{xx}(\widehat{X}^{i,*}_{t},t)\widehat{X}^{i,*}_{t}}\Big{)},$
(3.3) $\displaystyle c^{i,*}_{t}$
$\displaystyle=\dfrac{(V^{i}_{x})^{-1}\Big{(}U^{i}_{x}(\widehat{X}^{i,*}_{t},t)\big{(}\tilde{c}^{(-i)}_{t}\big{)}^{\theta_{i}},t\Big{)}\big{(}\tilde{c}^{(-i)}_{t}\big{)}^{\theta_{i}}}{\widehat{X}^{i,*}_{t}}.$
(3.4)

where $\widehat{X}^{i,*}$ solves (2.6) with $(\pi^{i,*},c^{i,*})$ being used.

Then, in the sense of Definition 3.1, if
$(\pi^{i,*},c^{i,*})\in\mathcal{A}^{i}$ and if $\widehat{X}^{i,*}$ is well-
defined, then $Q^{i}(x,t)$ is a forward relative performance process for
manager $i$ and, moreover, the policy $(\pi^{i,*},c^{i,*})$ is optimal.

Let us recall the concept of a CRRA utility map. Following e.g., [52, Section
5], we say a utility map $U$ is of Constant Relative Risk Aversion (CRRA) type
if the _local risk tolerance function_
$r:\Omega\times\mathbb{R}\times[0,\infty)\to\mathbb{R}$, given by the quotient
$r(x,t)=-U_{x}(x,t)/U_{xx}(x,t)$ is linear in space, i.e. $r(x,t)=\delta x,$
for any $\delta>0$ and $t\geqslant 0,\leavevmode\nobreak\ x\in\mathbb{R}$.
This is the case for the classical power utility function, see Section 3.3
below.

By direct inspection of the expression for $\pi^{i,*}$ in (3.3), if the local
risk tolerance function $r^{i}(x,t)=\delta_{i}x$, for any $t>0$ (e.g. CRRA
utilities) then the optimal investment strategy will be constant throughout
time if additionally all other agents also choose a constant investment
strategy.

######  Corollary 3.5 (Constant investment strategies under CRRA).

Assume $U^{i}$ and $V^{i}$ to be of CRRA type, i.e., the _local risk tolerance
function_ is linear in space uniformly. Assume further that all agents $j\neq
i$ invest according to constant investment strategies $\pi^{j}\in\mathbb{R}$.
Then, $\pi^{i,*}$ is constant.

At this point it is not clear that if the agents $j\neq i$ use deterministic
continuous consumption strategies $c^{j}$ then the optimal consumption
strategy $c^{i,*}$ in (3.4) is also so. We next prove the “best responses”
result, Proposition 3.4.

###### Proof of Proposition 3.4.

From (2.4) we have access to the dynamics of $d\widehat{X}^{i}$ and of
$d(X^{i}_{t}\big{(}\overline{X}^{(-i)}_{t}\big{)}^{-\theta_{i}})$. Using
(3.1), we apply the Itô formula to
$Q^{i}(\widehat{X}^{i}_{t},t)=Q^{i}\big{(}X^{i}_{t}\big{(}\overline{X}^{(-i)}_{t}\big{)}^{-\theta_{i}},t\big{)}$,
and obtain using the notation set before (2.6)

$\displaystyle dQ^{i}(\widehat{X}^{i}_{t},t)$
$\displaystyle=U^{i}_{t}(\widehat{X}^{i}_{t},t)dt+U^{i}_{x}(\widehat{X}^{i}_{t},t)d\widehat{X}^{i}_{t}+\frac{1}{2}U^{i}_{xx}(\widehat{X}^{i}_{t},t)d\langle\widehat{X}^{i}_{t}\rangle+V^{i}(\hat{c}^{i}_{t}\widehat{X}_{t}^{i},t)dt$
$\displaystyle=U^{i}_{t}(\widehat{X}^{i}_{t},t)dt+U^{i}_{x}(\widehat{X}^{i}_{t},t)\bigg{(}\mu_{i}\pi^{i}_{t}-\theta_{i}\overline{\mu\pi}^{(-i)}+\frac{\theta_{i}}{2}\overline{\Sigma\pi^{2}}^{(-i)}$
$\displaystyle\qquad+\frac{\theta_{i}^{2}}{2}\Big{(}\big{(}\overline{\sigma\pi}^{(-i)}\big{)}^{2}+\frac{1}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\Big{)}-\theta_{i}\sigma_{i}\pi^{i}_{t}\overline{\sigma\pi}^{(-i)}\bigg{)}\overline{X}^{i}_{t}dt$
$\displaystyle\qquad+U^{i}_{x}(\widehat{X}^{i}_{t},t)\big{(}\sigma_{i}\pi_{t}^{i}-\theta_{i}\overline{\sigma\pi}^{(-i)}\big{)}\widehat{X}^{i}_{t}dB_{t}$
(3.5)
$\displaystyle\qquad+U^{i}_{x}(\widehat{X}^{i}_{t},t)\widehat{X}^{i}_{t}\bigg{(}\nu_{i}\pi^{i}_{t}dW^{i}_{t}-\theta_{i}\Big{(}\frac{1}{n-1}\sum_{k\neq
i}^{n}\nu_{k}\pi^{k}_{t}dW^{k}_{t}\Big{)}\bigg{)}$
$\displaystyle\qquad+\frac{1}{2}U^{i}_{xx}(\widehat{X}^{i}_{t},t)\bigg{(}(\nu_{i}\pi^{i}_{t})^{2}+\frac{\theta_{i}^{2}}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}+\Big{(}\sigma_{i}\pi_{t}^{i}-\theta_{i}\overline{\sigma\pi}^{(-i)}\Big{)}^{2}\bigg{)}\big{(}\widehat{X}^{i}_{t}\big{)}^{2}dt,$
$\displaystyle\qquad-U^{i}_{x}(\widehat{X}^{i}_{t},t)\big{(}c^{i}_{t}-\theta_{i}\bar{c}^{(-i)}_{t}\big{)}\widehat{X}_{t}^{i}dt+V^{i}(\hat{c}^{i}_{t}\widehat{X}_{t}^{i},t)dt,$

with
$U^{i}(\widehat{X}^{i}_{0},0)=U^{i}\big{(}x^{i}_{0}(\overline{x}_{0}^{(-i)})^{-\theta_{i}},0\big{)}=u^{i}_{0}\big{(}x^{i}_{0}(\overline{x}_{0}^{(-i)})^{-\theta_{i}}\big{)}$
and we used that the $B,W^{j}$ are all i.i.d.

By Definition 3.1, the process $Q^{i}(\widehat{X}^{i}_{t},t)$ becomes a
martingale at the optimal $(\pi^{i,*},c^{i,*})$, hence, direct computations
using first order conditions
($\partial_{\pi^{i}}\textrm{``drift''}=\partial_{c^{i}}\textrm{``drift''}=0$)
yield

$\displaystyle\begin{cases}U^{i}_{x}\big{(}\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}\big{)}\widehat{X}_{t}^{i}+\frac{1}{2}U_{xx}^{i}\Big{[}2\pi^{i}_{t}\nu_{i}^{2}+2\big{(}\sigma_{i}\pi_{t}^{i}-\theta_{i}\overline{\sigma\pi}^{(-i)}\big{)}\sigma_{i}\Big{]}(\widehat{X}_{t}^{i})^{2}=0,\\\
-U^{i}_{x}(\widehat{X}_{t}^{i},t)\widehat{X}_{t}^{i}+V^{i}_{x}(\hat{c}^{i}_{t}\widehat{X}_{t}^{i},t)\frac{\widehat{X}_{t}^{i}}{\big{(}\tilde{c}^{(-i)}_{t}\big{)}^{\theta_{i}}}=0,\end{cases}$
$\displaystyle\qquad\Leftrightarrow\quad\begin{cases}\pi^{i}_{t}&=\dfrac{1}{\nu_{i}^{2}+\sigma_{i}^{2}}\Big{(}{\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}}-\big{(}\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}\big{)}\frac{U^{i}_{x}(\widehat{X}^{i}_{t},t)}{U^{i}_{xx}(\widehat{X}^{i}_{t},t)\widehat{X}^{i}_{t}}\Big{)},\\\
c^{i}_{t}&=\dfrac{(V^{i}_{x})^{-1}\Big{(}U^{i}_{x}(\widehat{X}^{i}_{t},t)\big{(}\tilde{c}^{(-i)}_{t}\big{)}^{\theta_{i}},t\Big{)}\big{(}\tilde{c}^{(-i)}_{t}\big{)}^{\theta_{i}}}{\widehat{X}_{t}^{i}}.\end{cases}$
(3.6)

Injecting $\pi^{i}_{t}$ in the drift term of (3.2) and simplifying we arrive
at the consistency condition (3.4), we do not carry out this step explicitly,
nonetheless, using that the pair $(U^{i},V^{i})$ solves (3.4), equation (3.2)
simplifies to

$\displaystyle dQ^{i}(\widehat{X}^{i}_{t},t)$
$\displaystyle=U^{i}_{x}(\widehat{X}^{i}_{t},t)\Big{(}\nu_{i}\pi_{t}^{i}dW_{t}^{i}-\theta_{i}\big{(}\frac{1}{n-1}\sum_{k\neq
i}^{n}\pi_{t}^{k}\nu_{k}dW_{t}^{k}\big{)}\Big{)}\widehat{X}^{i}_{t}+U^{i}_{x}(\widehat{X}^{i}_{t},t)\Big{(}\sigma_{i}\pi_{t}^{i}-\theta_{i}\overline{(\pi\sigma)}^{(-i)}_{t}\Big{)}\widehat{X}^{i}_{t}dB_{t}$
$\displaystyle\qquad+\frac{1}{2}U^{i}_{xx}(\widehat{X}^{i}_{t},t)\frac{1}{\nu_{i}^{2}+\sigma_{i}^{2}}\bigg{|}{\pi^{i}_{t}}(\nu_{i}^{2}+\sigma_{i}^{2})-\Big{(}\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}-\mu_{i}\frac{U^{i}_{x}(\widehat{X}^{i}_{t},t)}{U^{i}_{xx}(\widehat{X}^{i}_{t},t)\widehat{X}^{i}_{t}}\Big{)}\bigg{|}^{2}(\widehat{X}^{i}_{t})^{2}dt$
(3.7)
$\displaystyle\qquad+V^{i}(\hat{c}^{i}_{t}\widehat{X}_{t}^{i},t)dt-U^{i}_{x}(\widehat{X}^{i}_{t},t)c^{i}_{t}\widehat{X}_{t}^{i}dt-\widetilde{V}^{i}(U^{i}_{x}(\widehat{X}_{t}^{i},t),t)dt.$

The concavity assumption of $U^{i}(x,t)$ implies that the second line of the
term above is non-positive and vanishes when (3.2) holds. At the same time by
the definition of $\widetilde{V}^{i}$, the Fenchel-Legendre transform of
$V^{i}$, we notice that

$\widetilde{V}^{i}(U^{i}_{x}(\widehat{X}^{i}_{t},t),t)=V^{i}(\hat{c}^{i,*}_{t}\widehat{X}_{t}^{i},t)-U^{i}_{x}(\widehat{X}_{t}^{i},t)c^{i,*}_{t}\widehat{X}^{i}_{t},$

that yields non-positivity and extremality of the last line of (3.2), when
$c^{i}=c^{i,*}$. We can conclude that, if
$(\pi^{i,*}_{t},c^{i,*}_{t})=(\pi^{i}_{t},c^{i}_{t})\in\mathcal{A}^{i}$ and
the associated process $\widehat{X}^{i,*}$ is well-defined (solution to (2.6)
with $(\pi^{i,*},c^{i,*})$), the process $U^{i}(\widehat{X}^{i,*}_{t},t)$ is a
local-martingale, otherwise it is a local supermartingale. The result
concludes. ∎

Here in contrast with [46] we do not know the explicit form of the utility, it
being part of the solution, hence we exploit a convex duality to prove the
extremality argument and combine with HJB type methodology. In [34] or [22]
the authors only explored the investment problem and hence the above issue
does not appear.

###  3.3 Forward performance with initial CRRA power preference

For $i\in\\{1,\dots,n\\}$, let the dynamics of $Q^{i}$ be given by (3.1). Then
we find one part of the solution to PDE (3.4) as

$\displaystyle U^{i}(x,t)$
$\displaystyle=\begin{cases}\frac{1}{1-\frac{1}{\delta_{i}}}x^{1-\frac{1}{\delta_{i}}}f_{i}(t),&\delta\neq
1,\leavevmode\nobreak\ \delta>0,\\\
\log(x)f_{i}(t)+h_{i}(t),&\delta=1,\end{cases}$ (3.8)

where $(f_{i}(t))_{t\geqslant 0},(h_{i}(t))_{t\geqslant 0}$ are the random
maps independent of $x$ satisfying $f_{i}(0)=1$ and $h_{i}(0)=0$, are
sufficiently integrable and $t\mapsto f_{i}(t),\leavevmode\nobreak\ t\mapsto
h_{i}(t)$ are differentiable.

######  Remark 3.6.

For the sake of simplicity we omit the presentation for the logarithmic
utility. At the same time we emphasise that the optimal policy for this case
coincides with the general power case with $\delta_{i}=1$ plugged inside.

At the same time, the structure of the consistency condition (3.4) yields, see
[26, Proposition 4.5], the form of $V^{i}$

$\displaystyle V^{i}(x,t)$
$\displaystyle=\frac{1}{\epsilon_{i}}\frac{1}{1-\frac{1}{\delta_{i}}}x^{1-\frac{1}{\delta_{i}}}g_{i}(t),$
(3.9)

where $(g_{i}(t))_{t\geqslant 0}$ is the random map independent of $x$
satisfying $g_{i}(0)=1$ and sufficiently integrable. We refer to the constants
$\delta_{i}>0,\leavevmode\nobreak\ \delta_{i}\neq 1,\leavevmode\nobreak\
\theta_{i}\in[0,1]$ and $\epsilon_{i}>0$ as _personal risk tolerance_ and
_competition weight_ , respectively, whereas $\epsilon_{i}>0$ captures the
_relative importance the agent assigns to the wealth compared to consumption_.

In this case of initial power preferences we have, as before, that the _local
risk tolerance function_ for $U^{i}$ satisfies
$r^{i}=-U^{i}_{x}/U^{i}_{xx}=\delta_{i}x$, and hence

$\displaystyle\pi^{i,*}_{t}$
$\displaystyle=\frac{1}{\nu_{i}^{2}+\sigma_{i}^{2}}\Big{(}{\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}}+(\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)})\delta_{i}\Big{)}.$

We now proceed to find this case’s optimal consumption map $c^{i}$ and
utilities fields. Injecting the expressions above for
$U^{i}(x,t),\leavevmode\nobreak\ V^{i}(x,t)$ in (3.4) together with optimal
consumption given by (3.2) yields an ODE system for $(f_{i},g_{i},c^{i})$, we
have

$\displaystyle\begin{cases}c^{i}_{t}=\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(1-\delta_{i})}\epsilon_{i}^{-\delta_{i}}\bigg{(}\dfrac{g_{i}(t)}{f_{i}(t)}\bigg{)}^{\delta_{i}},\\\
f^{\prime}_{i}(t)+\Big{(}\rho_{i}+\big{(}1-\frac{1}{\delta_{i}}\big{)}\theta_{i}\bar{c}_{t}^{(-i)}\Big{)}f_{i}(t)+\frac{\epsilon_{i}^{-\delta_{i}}}{\delta_{i}}\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(1-\delta_{i})}f_{i}(t)\bigg{(}\dfrac{g_{i}(t)}{f_{i}(t)}\bigg{)}^{\delta_{i}}=0,\end{cases}$
(3.10)

where

$\displaystyle\rho^{i}_{t}$
$\displaystyle:=-\Big{(}1-\frac{1}{\delta_{i}}\Big{)}\bigg{(}\theta_{i}\overline{\mu\pi}^{(-i)}-\frac{\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}(\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)})}{\nu_{i}^{2}+\sigma_{i}^{2}}-\frac{\theta_{i}}{2}\overline{\Sigma\pi^{2}}^{(-i)}$
$\displaystyle\qquad\qquad-\frac{\theta_{i}^{2}}{2}\Big{(}\big{(}\overline{\sigma\pi}^{(-i)}\big{)}^{2}+\frac{1}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\Big{)}-\delta_{i}\frac{(\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)})^{2}}{2(\nu_{i}^{2}+\sigma_{i}^{2})}$
(3.11)
$\displaystyle\qquad\qquad-\frac{1}{2\delta_{i}}\Big{(}\Big{(}\theta_{i}\overline{\sigma\pi}^{(-i)}\Big{)}^{2}\Big{(}\frac{\sigma_{i}^{2}}{\nu_{i}^{2}+\sigma_{i}^{2}}-1\Big{)}-\frac{\theta_{i}^{2}}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\Big{)}\bigg{)}.$

We have two equations for three unknowns, now we need one further assumption
for the nature of $f_{i}$ and $g_{i}$ in order to solve the system.

######  Assumption 3.7.

Let $g_{i}(t)=f_{i}(t)^{\kappa_{i}}$, where $\kappa_{i}\in\mathbb{R}$.

######  Remark 3.8.

Here, as opposed to [46] (also [44, 60]) where one is naturally led to
$\kappa=0$, we have a non-trivial time-dependence of the consumption utility.
At this point we would like to highlight the FPP approach naturally supports
the time dynamics of the utilities, in contrast with the HJB approach, that
does not additionally weight the integrand with a function of time.

We have two distinct cases.

_Case 1: Let $\kappa_{i}\neq 1$._ We obtain and solve a classical ODE of
Bernoulli equation type (we omit the dependence in $t$ for simplification)

$\displaystyle
f^{\prime}_{i}+a_{i}(t)f_{i}+b_{i}(t)f_{i}^{1-(1-\kappa_{i})\delta_{i}}=0\quad\textrm{where}\quad
a_{i}(t):=\rho_{t}^{i}-\theta_{i}\bar{c}^{(-i)}_{t}\ \textrm{ and }\
b_{i}(t):=\frac{\epsilon_{i}^{-\delta_{i}}}{\delta_{i}}\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(1-\delta_{i})}.$

Substituting $k_{i}(t)=f_{i}(t)^{(1-\kappa_{i})\delta_{i}}$ and we find the
ODE for $k_{i}(t)$

$\displaystyle\frac{1}{(1-\kappa_{i})\delta_{i}}k_{i}^{\prime}+a_{i}(t)k_{i}+b_{i}(t)=0,$

Hence, solving the ODE we have

$\displaystyle
k_{i}(t)=e^{-(1-\kappa_{i})\delta_{i}\int_{0}^{t}a_{i}(s)ds}-(1-\kappa_{i})\delta_{i}\int_{0}^{t}e^{-(1-\kappa_{i})\delta_{i}\int_{s}^{t}a(r)dr}b_{i}(s)ds.$

and $f_{i}(t)=k_{i}(t)^{\frac{1}{(1-\kappa_{i})\delta_{i}}}$,
$g_{i}(t)=\frac{1}{\epsilon_{i}}f_{i}(t)^{\kappa_{i}}=\frac{1}{\epsilon_{i}}k_{i}(t)^{\frac{\kappa_{i}}{(1-\kappa_{i})\delta_{i}}}$.
Plugging the expressions for $f_{i}$ and $g_{i}$ in the first equation of
(3.10), the general form of the consumption strategy, we can extract the form
of optimal consumption map.

_Case 2. Let $\kappa_{i}=1$._ A strong simplification occurs and we have by
direct injection in (3.10)

$\displaystyle
c^{i}_{t}=\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(1-\delta_{i})}\epsilon_{i}^{\delta_{i}}.$

###  3.4 The Forward Nash equilibrium

In view of the _best responses_ of Proposition 3.4 we now solve for the
_simultaneous best responses_ as to establish the existence of a Nash
equilibrium.

######  Definition 3.9 (Forward Nash equilibrium).

Define the tuple of $n-1$ strategy pairs
($\pi^{*,-i},c^{*,-i}):=\big{(}(\pi^{1,*},c^{1,*}),\dots,(\pi^{i-1,*},c^{i-1,*}),(\pi^{i+1,*},c^{i+1,*}),\dots,(\pi^{n,*},c^{n,*})\big{)}$,
where for any $i\in\\{1,\dots,n\\}$ we have
$(\pi^{i,*},c^{i,*})\in\mathcal{A}^{i}$. Let for
$\big{(}\pi^{i},c^{i}\big{)}\in\mathcal{A}^{i}$. Let $Q^{i}$ be the
$\mathbb{F}$-progressively measurable random field satisfying

$Q^{i}(x,t):=U^{i}(x,t)+\int_{0}^{t}V^{i}(\hat{c}_{s}^{i}x,s)ds,$

where $\hat{c}^{i}$ is given by (2.5) with $c^{j}=c^{j,*},\leavevmode\nobreak\
j\neq i$ and $U^{i},V^{i}:\Omega\times\mathbb{R}\times[0,\infty)\to\mathbb{R}$
are two other $\mathbb{F}$-progressively measurable random fields.

A forward Nash equilibrium consists of $n$-triples of
$\mathcal{F}_{t}$-adapted maps $(Q^{i},\pi^{i,*},c^{i,*})$ with $i=1,\dots,n$
such that for any $t\geqslant 0$ the following conditions hold.

  1. 1.

The mappings $x\mapsto U^{i}(x,t)$ and $y\mapsto V^{i}(y,t)$ are
$\mathbb{P}$-a.s. strictly increasing and strictly concave;

  2. 2.

$Q^{i}(\widehat{X}^{i}_{t}(\pi^{*,-i},c^{*,-i}),t)$ is a (local)
supermartingale for $\widehat{X}^{i}(\pi^{*,-i},c^{*,-i})$ and $\hat{c}^{i,*}$
given by (2.6) and (2.5) respectively with all managers $j\neq i$ acting
according to $(\pi^{j,*},c^{j,*})$;

  3. 3.

$Q^{i}(\widehat{X}^{i,*}_{t}(\pi^{*,-i},c^{*,-i}),t)$ is a (local) martingale
for $\widehat{X}^{i}(\pi^{*,-i},c^{*,-i})$ and $\hat{c}^{i,*}$ given by (2.6)
and (2.5) respectively with _all_ managers $j$ acting according to
$(\pi^{j,*},c^{j,*})$.

If for all $j\in\\{1,\dots,n\\}$, $\pi^{j}_{t}=\pi^{j}$ are constant and
$c_{t}^{j}$ is continuous and deterministic for any $t\geqslant
0,\leavevmode\nobreak\ \mathbb{P}$-a.s., we say we have a _strong forward Nash
equilibrium_.

Under appropriate integrability conditions plus the martingale/supermartingale
characterisations, we have for some agent $i$, for any
$(\pi^{i},c^{i})\in\mathcal{A}^{i}$

$\displaystyle\mathbb{E}[Q^{i}(\widehat{X}^{i,*}_{t}(\pi^{*,-i},c^{*,-i}),t)]$
$\displaystyle=\mathbb{E}[Q^{i}(\widehat{X}^{i,*}_{0}(\pi^{*,-i},c^{*,-i}),0)]$
$\displaystyle=\mathbb{E}[Q^{i}(x^{i}_{0}(\overline{x}^{(-i)}_{0})^{-\theta_{i}},0)]$
$\displaystyle=Q^{i}(x^{i}_{0}(\overline{x}^{(-i)}_{0})^{-\theta_{i}},0)\geqslant\mathbb{E}[Q^{i}(\widehat{X}^{i}_{t}(\pi^{*,-i},c^{*,-i}),t)].$

As expected, no manager can increase the expected utility of her relative
performance metric by unilateral decision.

The solvability of the general forward Nash equilibrium seems very difficult
for a general forward criteria as one needs to solve the following system for
the $\pi^{i,*},c^{i,*}$ (see Proposition 3.4, in particular (3.2)) and the
corresponding PDEs with random coefficients for $U^{i}$ and $V^{i}$:

$\displaystyle\begin{cases}\pi^{i,*}_{t}(\nu_{i}^{2}+\sigma_{i}^{2})=\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}-\big{(}\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}\big{)}\dfrac{U^{i}_{x}(\widehat{X}^{i,*}_{t},t)}{U^{i}_{xx}(\widehat{X}^{i,*}_{t},t)\widehat{X}^{i,*}_{t}},\\\
-U^{i}_{x}(\widehat{X}_{t}^{i,*},t)\widehat{X}_{t}^{i,*}+V^{i}_{x}\Big{(}\frac{c^{i,*}_{t}}{\big{(}\tilde{c}^{(-i)}_{t}\big{)}^{\theta_{i}}}\widehat{X}_{t}^{i,*},t\Big{)}\frac{\widehat{X}_{t}^{i,*}}{\big{(}\tilde{c}^{(-i)}_{t}\big{)}^{\theta_{i}}}=0.\end{cases}$
(3.12)

### Equilibrium with FPPs of separable power factor form

In order to obtain explicit results we focus on the separable power factor
form case of (3.8)-(3.9) for which
$U^{i}_{x}/U^{i}_{xx}=V^{i}_{x}/V^{i}_{xx}=-\delta_{i}x$. More notably, at the
level at which we have formulated our problem we recover the results of [46,
Theorem 3] for which one has $U^{i}_{x}/U^{i}_{xx}=-\delta_{i}x$, for all $t$
(note their Remark 5).

######  Assumption 3.10.

Set $g_{i}(t):=f_{i}(t)^{\kappa}$ in (3.8)-(3.9) for some
$\kappa\in\mathbb{R}$ for any $i=1,\dots,n$.

One can see that the PDE given by (3.4) depends on the two unknown functions
$U$ and $V$, and thus makes for an under-determined system. Hence, Assumption
3.10 is essential to ensure that the equation is solvable for the unique Nash
equilibrium to be given later in (3.14)–(3.15). At the same time we have to
assume the $\kappa$ to be constant among the whole population in order to get
a solution for the Nash equilibria.

######  Theorem 3.11.

Let the conditions of Proposition 3.4 hold for all agents
$i\in\\{1,\dots,n\\}$. Assume furthermore that agents have separable power
factor form forward maps $U^{i},V^{i}$ with initial conditions.

$\displaystyle
U^{i}(0,x)=\epsilon_{i}V^{i}(0,x)=\frac{1}{1-\frac{1}{\delta_{i}}}x^{1-\frac{1}{\delta_{i}}},\quad\epsilon_{i}>0,\
\delta_{i}>0,\ \delta_{i}\neq 1,$

as in (3.8)-(3.9). Define the quantities

$\displaystyle\varphi^{\sigma}_{n}:=\frac{1}{n}\sum_{k=1}^{n}\delta_{k}\frac{\sigma_{k}\mu_{k}}{{\nu_{k}^{2}+\sigma_{k}^{2}\big{(}1+\frac{\theta_{k}(1-\delta_{k})}{n-1}\big{)}}},\qquad\psi^{\sigma}_{n}:=\frac{1}{n-1}\sum_{i=k}^{n}\theta_{k}(1-\delta_{k})\frac{\sigma^{2}_{k}}{{\nu_{i}^{2}+\sigma_{i}^{2}\big{(}1+\frac{\theta_{k}(1-\delta_{k})}{n-1}\big{)}}},$
$\displaystyle\lambda_{i}=\epsilon_{i}^{-\frac{\delta_{i}}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}}\Big{(}\widetilde{\epsilon^{\delta}}\Big{)}^{\frac{\theta_{i}(1-\delta_{i})}{{(\overline{\theta(1-\delta)}-1)\big{(}1+\frac{\theta_{i}}{n-1}(1-\delta_{i})\big{)}}}},\quad\beta_{i}=\frac{1}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}\bigg{(}\frac{\theta_{i}(1-\delta_{i})}{\overline{\theta(1-\delta)}-1}\overline{\rho\delta}-{\rho_{i}\delta_{i}}\bigg{)},$
$\displaystyle\widetilde{\epsilon^{\delta}}=\Bigg{(}\prod_{k=1}^{n}\epsilon_{k}^{\frac{\delta_{k}}{1+\frac{\theta_{k}}{n-1}(1-\delta_{k})}}\Bigg{)}^{\frac{1}{n}},\quad\overline{\rho\delta}=\frac{1}{n}\sum_{k=1}^{n}\frac{\rho_{k}\delta_{k}}{1+\frac{\theta_{k}}{n-1}(1-\delta_{k})},\quad\overline{\theta(1-\delta)}=\frac{1}{n-1}\sum_{k=1}^{n}\frac{\theta_{k}(1-\delta_{k})}{1+\frac{\theta_{k}}{n-1}(1-\delta_{k})}.$

Let as well, for $i=1,\dots,n$, the map $\rho^{i}_{t}$ be defined as

$\displaystyle\rho^{i}_{t}=\rho_{i}$
$\displaystyle:=-\Big{(}1-\frac{1}{\delta_{i}}\Big{)}\bigg{(}\theta_{i}\overline{\mu\pi}^{(-i)}-\frac{\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)}(\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)})}{\nu_{i}^{2}+\sigma_{i}^{2}}-\frac{\theta_{i}}{2}\overline{\Sigma\pi^{2}}^{(-i)}$
$\displaystyle\qquad\qquad-\frac{\theta_{i}^{2}}{2}\Big{(}\big{(}\overline{\sigma\pi}^{(-i)}\big{)}^{2}+\frac{1}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\Big{)}-\delta_{i}\frac{(\mu_{i}-\theta_{i}\sigma_{i}\overline{\sigma\pi}^{(-i)})^{2}}{2(\nu_{i}^{2}+\sigma_{i}^{2})}$
(3.13)
$\displaystyle\qquad\qquad-\frac{1}{2\delta_{i}}\Big{(}\Big{(}\theta_{i}\overline{\sigma\pi}^{(-i)}\Big{)}^{2}\Big{(}\frac{\sigma_{i}^{2}}{\nu_{i}^{2}+\sigma_{i}^{2}}-1\Big{)}-\frac{\theta_{i}^{2}}{n-1}\overline{(\nu\pi)^{2}}^{(-i)}\Big{)}\bigg{)}.$

where the relevant expressions for
$\overline{\sigma\pi}^{(-i)},\leavevmode\nobreak\
\overline{\mu\pi}^{(-i)},\leavevmode\nobreak\ \overline{(\nu\pi)^{2}}^{(-i)}$
and $\overline{\Sigma\pi^{2}}^{(-i)}$ are given below in (3.16), (3.17),
(3.18), (3.19) respectively.

If $\psi^{\sigma}_{n}\neq 1$ and Assumption 3.10 holds ($\kappa_{i}=\kappa$
for all $i$) then a unique _forward Nash equilibrium_ exists with the optimal
strategies $\pi^{i,*}$ and $c^{i,*}$ given by

$\displaystyle\pi^{i,*}$
$\displaystyle=\frac{1}{{\nu_{i}^{2}+\sigma_{i}^{2}\big{(}1+\frac{\theta_{i}(1-\delta_{i})}{n-1}\big{)}}}\Big{(}\theta_{i}\sigma_{i}(1-\delta_{i})\Big{(}1+\frac{1}{n-1}\Big{)}\frac{\varphi^{\sigma}_{n}}{1-\psi^{\sigma}_{n}}+\mu_{i}\delta_{i}\Big{)},$
(3.14) $\displaystyle c^{i,*}_{t}$
$\displaystyle=\begin{cases}\bigg{(}\frac{1}{\beta_{i}}+\Big{(}\frac{1}{\lambda_{i}}-\frac{1}{\beta_{i}}\Big{)}e^{-(\kappa-1)\beta_{i}t}\bigg{)}^{-1},&\beta_{i}\neq
0,\\\
\big{(}(\kappa-1)t+\frac{1}{\lambda_{i}}\big{)}^{-1},&\beta_{i}=0,\end{cases}$
(3.15)

and $\rho^{i}_{t}$ in (3.11) is constant in time. Moreover, the Nash
equilibrium strategy is strong if $\kappa>1$ and
$\kappa<1,\leavevmode\nobreak\ \beta_{i}>\lambda_{i}$.

The forward Nash equilibria is given by the $n$-triples
$\\{(Q^{i,*},\pi^{i,*},c^{i,*})\\}_{i=1,\dots,n}$ where the
$Q^{i,*}(x,t)=U^{i,*}(x,t)+\int_{0}^{t}V^{i,*}(\hat{c}^{i}_{s}x,s)ds$, with
$U^{i,*},V^{i,*}$ the solution of (3.4) (of the form (3.8)-(3.9)) with the
optimal strategies $\pi^{\cdot,*},c^{\cdot,*}$ plugged-in.

In Section 5 the parameter $\kappa$ is interpreted as market consumption
intensity and we discuss at length its economic features and comparative
interpretation in Section 5 below. The investment strategy (3.14) is a
classical expression matching that of [47, 46, 22]. The expression for the
consumption strategy (3.15) has similarities to the results found in [46,
Equation (5)] with the crucial difference from $\kappa$. If $\kappa=0$ then
the results coincide with [46].

###### Proof.

_Step 1. Finding the investment strategy._

First, we deal with the investment policy. Injecting the condition
$U_{x}/U_{xx}=-\delta_{i}x$ in (3.12), the system to be solved in order to
ascertain the Nash equilibrium is ($i\in\\{1,\dots,n\\}$)

$\displaystyle\pi^{i,*}_{t}(\nu_{i}^{2}+\sigma_{i}^{2})$
$\displaystyle=\theta_{i}\sigma_{i}(1-\delta_{i})\Big{(}\frac{1}{n-1}\sum_{k=1,k\neq
i}^{n}\sigma_{k}\pi^{k,*}_{t}\Big{)}+\mu_{i}\delta_{i}$
$\displaystyle=\theta_{i}\sigma_{i}(1-\delta_{i})\Big{(}\frac{n}{n-1}\overline{\sigma\pi}-\frac{1}{n-1}\sigma_{i}\pi^{i,*}\Big{)}+\mu_{i}\delta_{i}$
$\displaystyle\Leftrightarrow\pi^{i,*}_{t}$
$\displaystyle=\frac{1}{{\nu_{i}^{2}+\sigma_{i}^{2}\big{(}1+\frac{\theta_{i}(1-\delta_{i})}{n-1}\big{)}}}\Big{(}\theta_{i}\sigma_{i}(1-\delta_{i})\frac{n}{n-1}\overline{\sigma\pi}+\mu_{i}\delta_{i}\Big{)}.$

The final line yields the expression for $\pi^{i,*}$ as a function of the
unknown $\overline{\sigma\pi}$. To determine the latter, multiply both sides
by $\sigma_{i}$ and average over $i\in\\{1,\dots,n\\}$, this yields a
solvability condition

$\displaystyle\overline{\sigma\pi}=\overline{\sigma\pi}\psi^{\sigma}_{n}+\varphi^{\sigma}_{n}\quad\Leftrightarrow\quad\overline{\sigma\pi}=\frac{\varphi^{\sigma}_{n}}{1-\psi^{\sigma}_{n}}\quad\textrm{as
long as }\quad\psi^{\sigma}_{n}\neq 1.$ (3.16)

Plugging the expression $\overline{\sigma\pi}$ in that for $\pi^{i,*}$ yields
the result. That the optimal strategies are constant is now obvious and one
finds that the corresponding $\rho$ (see (3.11)) is time-independent with
$\rho_{t}^{i}=\rho_{i}$.

_Step 2. Finding the explicit form of $\rho_{i}$._

Just like for $\overline{\sigma\pi}$, we obtain an expression for
$\overline{\mu\pi}$ by multiplying $\pi^{i,*}$ by $\mu_{i}$ and averaging on
both sides, we have

$\displaystyle\overline{\mu\pi}=\frac{n}{n-1}\frac{\varphi^{\sigma}_{n}}{1-\psi^{\sigma}_{n}}\psi^{\mu}_{n}+\phi^{\mu}_{n},\qquad\overline{\mu\pi}^{(-i)}=\frac{n}{n-1}\overline{\mu\pi}-\frac{1}{n-1}\mu_{i}\pi^{i}.$
(3.17)

where $\varphi^{\mu}_{n}$, $\psi^{\mu}_{n}$ are defined as

$\varphi^{\mu}_{n}:=\frac{1}{n}\sum_{k=1}^{n}\delta_{k}\frac{\mu^{2}_{k}}{\nu_{k}^{2}+\sigma_{k}^{2}(1+\frac{(1-\delta_{k})\theta_{k}}{n-1})},\qquad\psi^{\mu}_{n}:=\frac{1}{n-1}\sum_{k=1}^{n}\theta_{k}(1-\delta_{k})\frac{\mu_{k}\sigma_{k}}{\nu_{k}^{2}+\sigma_{k}^{2}(1+\frac{(1-\delta_{k})\theta_{k}}{n-1})}.$

Similarly, defining

$\displaystyle\overline{(\nu\pi)^{2}}:=\frac{1}{n-1}\sum_{k\neq
i}(\nu_{k}\pi^{k}_{t})^{2},\quad\textrm{we
have}\quad\overline{(\nu\pi)^{2}}=\frac{1}{n}\sum_{k=1}^{n}\Big{(}\dfrac{\nu_{k}\theta_{k}\sigma_{k}\cdot\frac{n}{n-1}\cdot\frac{\varphi^{\sigma}_{n}}{1-\psi^{\sigma}_{n}}+\nu_{k}\mu_{k}\delta_{k}}{{\nu_{k}^{2}+\sigma_{k}^{2}\big{(}1+\frac{\theta_{k}(1-\delta_{k})}{n-1}\big{)}}}\Big{)}^{2},$
(3.18)

together with
$\overline{(\nu\pi)^{2}}^{(-i)}=\frac{n}{n-1}\overline{(\nu\pi)^{2}}-\frac{1}{n-1}(\nu_{i}\pi^{i})^{2}$.
Finally
$\overline{\Sigma\pi^{2}}=\overline{(\nu\pi)^{2}}+\overline{(\sigma\pi)^{2}}$,
and like (3.18),

$\displaystyle\overline{(\sigma\pi)^{2}}=\frac{1}{n}\sum_{k=1}^{n}\bigg{(}\dfrac{\sigma_{k}\theta_{k}\sigma_{k}\frac{n}{n-1}\frac{\varphi^{\sigma}_{n}}{1-\psi^{\sigma}_{n}}+\sigma_{k}\mu_{k}\delta_{k}}{{\nu_{k}^{2}+\sigma_{k}^{2}\big{(}1+\frac{(1-\delta_{k})\theta_{k}}{n-1}\big{)}}}\bigg{)}^{2},\quad\text{with}\quad\widehat{\Sigma\pi^{2}}=\frac{n}{n-1}\overline{\Sigma\pi^{2}}-\frac{1}{n-1}(\nu_{i}^{2}+\sigma_{i}^{2})(\pi^{i})^{2}.$
(3.19)

Replacing these expressions in that for constant $\rho^{i}_{t}=\rho_{i}$ in
equation (3.3) from Section 3.3 the expression in the result’s statement
follows.

_Step 3. Finding the consumption strategy and relative performance utilities._

The system of equations (3.10) under Assumption 3.10, i.e.,
$g_{i}(t)=f_{i}(t)^{\kappa}$, becomes

$\displaystyle\begin{cases}c^{i}_{t}=\epsilon_{i}^{-\delta_{i}}\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(1-\delta_{i})}f_{i}(t)^{(\kappa-1)\delta_{i}},\\\
f^{\prime}_{i}(t)+\Big{(}\rho_{i}+\theta_{i}\big{(}1-\frac{1}{\delta_{i}}\big{)}\bar{c}_{t}^{(-i)}\Big{)}f_{i}(t)+\frac{\epsilon_{i}^{-\delta_{i}}}{\delta_{i}}\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(1-\delta_{i})}f_{i}(t)^{(\kappa-1)\delta_{i}+1}=0,\end{cases}$
(3.20)

where $\rho$ is given by (3.11). We mainly follow the machinery of the proof
[46, Theorem 2.2] to obtain the closed form solution. We repeat these
arguments, but emphasise the notable discrepancies. Substituting the LHS of
the first equation of (3.20) into the second, we obtain the linear ODE to
solve for

$\displaystyle
f_{i}(t)=\exp\Big{(}-\int_{0}^{t}\big{(}\rho_{i}+\theta_{i}\big{(}1-\frac{1}{\delta_{i}}\big{)}\bar{c}_{s}^{(-i)}+\frac{1}{\delta_{i}}c^{i}_{s}\big{)}ds\Big{)}.$
(3.21)

Now plugging it back in the first equation of (3.20) we obtain

$\displaystyle c^{i}_{t}\exp\Big{(}(\kappa-1)\int_{0}^{t}c^{i}_{s}ds\Big{)}$
$\displaystyle=\epsilon_{i}^{-\delta_{i}}\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(1-\delta_{i})}e^{(\kappa-1)\delta_{i}\rho_{i}t}\exp\Big{(}(\kappa-1)(1-\delta_{i})\theta_{i}\int_{0}^{t}\big{(}\bar{c}_{s}^{(-i)}\big{)}ds\Big{)}.$
(3.22)

After rewriting it with respect to
$\tilde{c}_{t}=\Big{(}\prod_{k=1}^{n}c^{k}_{t}\Big{)}^{\frac{1}{n}}$ and
$\bar{c}_{t}=\frac{1}{n}\sum_{k=1}^{n}c^{k}_{t}$, we get

$\displaystyle c^{i}_{t}\exp\Big{(}(\kappa-1)\int_{0}^{t}c^{i}_{s}ds\Big{)}$
$\displaystyle=\epsilon_{i}^{-\frac{\delta_{i}}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}}\tilde{c}_{t}^{\frac{n}{n-1}\frac{\theta_{i}(1-\delta_{i})}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}}e^{\frac{(\kappa-1)\rho_{i}\delta_{i}}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}t}$
(3.23)
$\displaystyle\qquad\qquad\times\exp\Big{(}(\kappa-1)\frac{n}{n-1}\frac{\theta_{i}(1-\delta_{i})}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}\int_{0}^{t}\bar{c}_{s}ds\Big{)}.$

We take the geometric average to the equation (3.23) over $i=1,\dots,N$ to get

$\displaystyle\tilde{c}_{t}\exp\Big{(}(\kappa-1)\int_{0}^{t}\bar{c}_{s}ds\Big{)}=\Big{(}\widetilde{\epsilon^{\delta}}\Big{)}^{-1}e^{(\kappa-1)\overline{\rho\delta}t}\bigg{(}\tilde{c}_{t}\exp\Big{(}(\kappa-1)\int_{0}^{t}\bar{c}_{s}ds\Big{)}\bigg{)}^{\overline{\theta(1-\delta)}},$
(3.24)

where

$\displaystyle\widetilde{\epsilon^{\delta}}=\Bigg{(}\prod_{k=1}^{n}\epsilon_{k}^{\frac{\delta_{k}}{1+\frac{\theta_{k}}{n-1}(1-\delta_{k})}}\Bigg{)}^{\frac{1}{n}},\quad\overline{\rho\delta}=\frac{1}{n}\sum_{k=1}^{n}\frac{\rho_{i}\delta_{i}}{1+\frac{\theta_{k}}{n-1}(1-\delta_{k})},\quad\overline{\theta(1-\delta)}=\frac{1}{n-1}\sum_{k=1}^{n}\frac{\theta_{k}(1-\delta_{k})}{1+\frac{\theta_{k}}{n-1}(1-\delta_{k})}.$

Hence we obtain that

$\displaystyle\tilde{c}_{t}\exp\Big{(}(\kappa-1)\int_{0}^{t}\bar{c}_{s}ds\Big{)}=\Big{(}\widetilde{\epsilon^{\delta}}\Big{)}^{\frac{1}{\overline{\theta(1-\delta)}-1}}e^{\frac{(\kappa-1)\overline{\rho\delta}}{\overline{\theta(1-\delta)}-1}t}.$
(3.25)

Using the previous equation we rewrite (3.23) for

$\displaystyle c^{i}_{t}\exp\Big{(}(\kappa-1)\int_{0}^{t}c^{i}_{s}ds\Big{)}$
$\displaystyle=\lambda_{i}e^{\beta_{i}(\kappa-1)t},$ (3.26)

where

$\displaystyle\lambda_{i}=\epsilon_{i}^{-\frac{\delta_{i}}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}}\Big{(}\widetilde{\epsilon^{\delta}}\Big{)}^{\frac{n}{n-1}\frac{\theta_{i}(1-\delta_{i})}{{(\overline{\theta(1-\delta)}-1)(1+\frac{\theta_{i}}{n-1}(1-\delta_{i}))}}},\quad\beta_{i}=\frac{1}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}\bigg{(}\frac{n}{n-1}\frac{\theta_{i}(1-\delta_{i})}{\overline{\theta(1-\delta)}-1}\overline{\rho\delta}-{\rho_{i}\delta_{i}}\bigg{)}.$

We now consider two distinct cases $\kappa\neq 1$ and $\kappa=1$.

_Case 1: Let $\kappa\neq 1$._ Integrating the (3.26), from $0$ to $t$ and
taking logarithm, we get

$\displaystyle(\kappa-1)\int_{0}^{t}c_{s}^{i}ds=\begin{cases}\log\Big{(}1+\frac{\lambda_{i}}{\beta_{i}}\big{(}e^{(\kappa-1)\beta_{i}t}-1\big{)}\Big{)},&\beta_{i}\neq
0,\\\ \log(1+\lambda_{i}(\kappa-1)t),&\beta_{i}=0.\end{cases}$ (3.27)

Finally differentiating (3.27) with respect to $t$, we obtain

$\displaystyle
c_{t}^{i}=\begin{cases}\bigg{(}\frac{1}{\beta_{i}}+\Big{(}\frac{1}{\lambda_{i}}-\frac{1}{\beta_{i}}\Big{)}e^{-(\kappa-1)\beta_{i}t}\bigg{)}^{-1},&\beta_{i}\neq
0,\\\
\big{(}(\kappa-1)t+\frac{1}{\lambda_{i}}\big{)}^{-1},&\beta_{i}=0.\end{cases}$
(3.28)

By direct inspection it is obvious, that for certain combinations of
parameters, namely $\kappa<1,\leavevmode\nobreak\ \beta_{i}<\lambda_{i}$ and
$\kappa<1,\leavevmode\nobreak\ \beta_{i}=0$, the optimal consumption $c$ in
not continuous and even can be negative. (however, the first case is
admissible, due to the exponential convergence of the integral), see Table 5.1
below for a summary.

Now from the first line of (3.20), we get that

$\displaystyle
f_{i}(t)=\Big{(}\big{(}c_{t}^{i}\big{)}^{\frac{1}{\delta_{i}}}\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(\frac{1}{\delta_{i}}-1)}\epsilon_{i}\Big{)}^{\frac{1}{\kappa-1}}\quad\text{and
thus}\quad
g_{i}(t)=\Big{(}\big{(}c_{t}^{i}\big{)}^{\frac{1}{\delta_{i}}}\big{(}\tilde{c}_{t}^{(-i)}\big{)}^{\theta_{i}(\frac{1}{\delta_{i}}-1)}\epsilon_{i}\Big{)}^{\frac{\kappa}{\kappa-1}}.$
(3.29)

_Case 2: Let $\kappa=1$._ Then (3.26) immediately yields
$c_{t}^{i}=\lambda_{i}$ which is the same result if one sets $\kappa=1$ in
(3.28).

∎

We comment further on the admissibility of $c^{*}$ in Section 5. It happens
that $c^{*}$ can be admissible but not form a strong equilibrium while
$\pi^{*}$ will always do so.

######  Remark 3.12 (Open question).

To obtain a Nash equilibria result we are restricted to $\kappa$ being
constant across the whole population of agents, and the same will happen for
the MFG case of next Section. The constraint is technical in nature and stems
from the difficulty in expressing the consistency equation for the averages of
all $c^{i}$ (i.e., (3.25) is not longer reachable) without which the system
cannot be solved. At the same time, for a “best responses" all the
$\kappa_{i}$’s may be distinct. The open question is then, if one allows for
different $\kappa_{i}$’s how to carry out the aggregation procedure as to
express the consistency equation for the averages of all $c^{i}$?

An example of how this problem was overcome can be found in [10]. There,
aggregation was achieved through so-called weighted-dilated inf-convolution.

Lastly, if one sets $\kappa=0$ then our FPP solution can be rewritten in
respect to a time horizon $T$ to recover exactly the results of [46] (and [47]
if $V^{i}=0$) and those of [34] for two players and no consumption. Such
calculations are straightforward hence omitted. We close the section with a
simplification corollary for the single stock case.

######  Corollary 3.13.

Let $\mu_{i}=\mu,\leavevmode\nobreak\ \sigma_{i}=\sigma,\leavevmode\nobreak\
\nu_{i}=0$, for all $i=1,\dots,n$ and $\mu,\sigma>0$. Then
$\lambda_{i},\beta_{i}$ are expressed as

$\displaystyle\lambda_{i}$
$\displaystyle=\epsilon_{i}^{-\frac{\delta_{i}}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}}\Big{(}\widetilde{\epsilon^{\delta}}\Big{)}^{\frac{\theta_{i}(1-\delta_{i})}{{(\overline{\theta(1-\delta)}-1)(1+\frac{\theta_{i}}{n-1}(1-\delta_{i}))}}},$
(3.30) $\displaystyle\beta_{i}$
$\displaystyle=\frac{\mu^{2}}{2\sigma^{2}\big{(}1+\frac{\theta_{i}}{n-1}(1-\delta_{i})\big{)}}\Big{(}1-\frac{\delta_{i}}{1+\frac{\theta_{i}}{n-1}(1-\delta_{i})}\Big{)}\Big{(}1-\frac{\theta_{i}}{\theta_{\textrm{crit}}\big{(}1+\frac{\theta_{i}}{n-1}\big{)}}\Big{)}\Big{(}\delta_{i}+\frac{\theta_{i}}{\theta_{\textrm{crit}}}(1-\delta_{i})\Big{)},$
(3.31)

where
$\theta_{\textrm{crit}}=\dfrac{1-\frac{1}{n-1}\sum_{k=1}^{n}\frac{\theta_{k}(1-\delta_{k})}{1+\frac{\theta_{k}}{n-1}(1-\delta_{k})}}{\frac{1}{n-1}\sum_{k=1}^{n}\frac{\delta_{k}}{1+\frac{\theta_{k}}{n-1}(1-\delta_{k})}}.$

Then, if $\theta_{\text{crit}}\neq 1$ then a _Nash equilibrium_ exists, with
the optimal strategy $(\pi^{i,*},c^{i,*})$ given by

$\displaystyle\pi^{i,*}$
$\displaystyle=\frac{\mu}{\sigma^{2}\big{(}1+\frac{\theta_{i}}{n-1}(1-\delta_{i})\big{)}}\Big{(}\delta_{i}+\frac{\theta_{i}}{\theta_{\textrm{crit}}}(1-\delta_{i})\Big{)},$
(3.32) $\displaystyle c^{i,*}_{t}$
$\displaystyle=\begin{cases}\bigg{(}\frac{1}{\beta}+\Big{(}\frac{1}{\lambda}-\frac{1}{\beta}\Big{)}e^{-(\kappa-1)\beta
t}\bigg{)}^{-1},&\beta\neq 0,\\\
\big{(}(\kappa-1)t+\frac{1}{\lambda}\big{)}^{-1},&\beta=0.\end{cases}$ (3.33)

Furthermore, the Nash equilibrium is strong, if either $\kappa>1$, or
$\kappa<1,\leavevmode\nobreak\ \lambda_{i}<\beta_{i}$.

##  4 The mean field forward optimisation game

By inspection of Theorem 3.11 one sees that the optimal strategies and FPP of
an agent depends on that agent’s specific parameters (model parameters,
initial wealth, risk tolerance and performance concern, etc.) and on certain
averages of the parameters of all agents. This motivates an MFG approach to
the game. In this section, and inspired by the results in the previous one and
[22], we formalise the concept of CRRA forward mean field Nash game. We use
the concept of _type distributions_ introduced in [47] and also used in [46,
22].

We focus on initial forward maps that at time $t=0$ are of power type,

$\displaystyle U^{i}(x,m,0)$
$\displaystyle=\epsilon_{i}V^{i}(x,m,0)=\begin{cases}\frac{1}{1-\frac{1}{\delta_{i}}}\big{(}xm^{-\theta_{i}}\big{)}^{1-\frac{1}{\delta_{i}}},&\delta_{i}\neq
1,\\\ \log(xm^{-\theta_{i}}),&\delta_{i}=1,\end{cases}$

where $x,m$ denote the wealth of agent and the average wealth of other agents
respectively (for $U^{i}$) or the consumption of the agent and the average
consumption of the other agents respectively (for $V^{i}$). We refer to
$\delta_{i}>0$ and $\theta_{i}\in[0,1]$ as _personal risk tolerance_ ,
_competition weight_ parameters, respectively, whereas $\epsilon_{i}>0$
captures the _relative importance the agent assigns to the wealth compared to
consumption_. Here, as in previous section (see Remark 3.6), we skip the
logarithmic utility case.

For the $n$-agent game, we define for each agent $i=1,\dots,n$ the _type
vector_

$\zeta_{i}:=(x^{i}_{0},\delta_{i},\theta_{i},\epsilon_{i},\mu_{i},\nu_{i},\sigma_{i}),$

which characterises uniquely each agent $i$. These _type vectors_ induce an
empirical measure, called the _type distribution_ , which is the probability
measure on the _type space_

$\displaystyle\mathcal{Z}^{e}:=\mathbb{R}\times(0,\infty)\times[0,1]\times(0,\infty)\times(0,\infty)\times[0,\infty)\times[0,\infty),$
(4.1)

given by

$m_{n}(A)=\frac{1}{n}\sum_{i=1}^{n}1_{A}(\zeta_{i}),\ \text{ for Borel sets
}A\subset\mathcal{Z}^{e}.$

Assume now that as the number of agents becomes large, $n\rightarrow\infty$,
the above empirical measure $m_{n}$ has a weak limit $m$, in the sense that
$\int_{\mathcal{Z}^{e}}f\,dm_{n}\rightarrow\int_{\mathcal{Z}^{e}}f\,dm$ for
every bounded continuous function $f$ on $\mathcal{Z}^{e}$. For example, this
holds almost surely if the $\zeta_{i}$’s are i.i.d. samples from $m$. Let
$\zeta=(\xi,\delta,\theta,\epsilon,\mu,\nu,\sigma)$ denote a an
$\mathcal{Z}^{e}$-valued random variable with this limiting distribution $m$.

The _mean field game_ (MFG) we define next allows us to derive the limiting
strategy as the outcome of a self-contained equilibrium problem, which
intuitively represents a game with a continuum of agents with type
distribution $m$. Rather than directly modelling a continuum of agents, we
follow the MFG paradigm of modelling a single generic agent, who we view as
randomly selected from the population. The probability measure $m$ represents
the distribution of type parameters among the continuum of agents;
equivalently, the generic agent’s type vector is a random variable with law
$m$. Heuristically, each agent in the continuum trades in a single stock
driven by two Brownian motions, one of which is unique to this agent and one
of which is common to all agents. We extend the Forward Nash equilibrium of
Definition 3.9 to the MFG setting below.

###  4.1 Agents as type-distribution samples and the market

Let $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F})_{t\geqslant 0},\mathbb{P})$
be a stochastic basis supporting two independent Brownian motions
$W=(W_{t})_{t\geqslant 0}$ and $B=(B_{t})_{t\geqslant 0}$ together with a
random vector $\zeta$ having distribution $m$ and given by

$\zeta=(\xi,\delta,\theta,\epsilon,\mu,\nu,\sigma),$

with values in the space $\mathcal{Z}^{e}$ defined in (4.1), and independent
of $W$ and $B$. Let $\mathbb{F}=(\mathcal{F}_{t})_{t\in[0,T]}$ denote the
smallest filtration satisfying the usual assumptions for which $\zeta$ is
$\mathcal{F}^{\mathrm{MF}}_{0}$-measurable and both $W$ and $B$ are adapted.
Let also $\mathbb{F}^{B}=(\mathcal{F}^{B}_{t})_{t\in[0,T]}$ denote the natural
filtration generated by the Brownian motion $B$.

The _generic agent’s_ wealth process solves

$\displaystyle dX_{t}=\pi_{t}X_{t}(\mu dt+\nu dW_{t}+\sigma
dB_{t})-c_{t}X_{t}dt,\quad X_{0}=\xi,$ (4.2)

where the pair $(\pi,c)$ must belong to the admissible set

$\displaystyle\mathcal{A}_{\mathrm{MF}}=\Big{\\{}(\pi,c)$
$\displaystyle:\textrm{$\mathbb{F}$-progressively measurable
$\mathbb{R}\times\mathbb{R}_{+}$-valued process }(\pi_{t},c_{t})_{t\geqslant
0},$ $\displaystyle\qquad\textrm{such that
}\mathbb{E}\Big{[}\int_{0}^{t}(|\pi_{s}|^{2}+|c_{s}|^{2})ds\Big{]}<\infty,\
\textrm{for any }t>0\Big{\\}}.$

The random variable $\xi$ is the initial wealth of the generic agent, whereas
$(\mu,\nu,\sigma)$ are the market parameters. In the sequel, the parameters
$\delta,\leavevmode\nobreak\ \theta$ and $\epsilon$ will affect the risk and
consumption preferences of the generic agent. Each agent among the continuum
will have different preference parameters and hence these seven parameters are
$\mathcal{F}_{0}$-random, and each has the exact same interpretation an in the
$n$-player game of the earlier section.

###  4.2 The mean field equilibrium

The formulation of the forward Nash game of Section 3 drives the formulation
of the mean field game we discuss here, see also [46, 22]. Recall that in the
MFG-formulation the _generic agent_ has no influence on the average wealth of
the continuum of agents, as but one agent amid a continuum.

We next introduce the concept of _mean field (MF)-forward relative
performance_ , $\pi^{*},c^{*}$ is the _MF-equilibrium_ and, the main object of
interest the _MF-Forward relative performance equilibrium_.

######  Assumption 4.1.

Assume that the derivatives $U_{t}(x,t),\leavevmode\nobreak\
U_{x}(x,t),\leavevmode\nobreak\ U_{xx}(x,t)$ and $V_{y}(y,t),V_{yy}(y,t)$
exist for all $t\geqslant 0,\leavevmode\nobreak\
x,y\in\mathbb{R},\leavevmode\nobreak\ \mathbb{P}$-a.s. and furthermore,
$x\mapsto U(x,t)$ and $y\mapsto V(y,t)$ are strictly increasing
($U_{x},V_{y}>0$) and strictly concave ($U_{xx},V_{yy}<0$) for any $t\geqslant
0,\leavevmode\nobreak\ x,y\in\mathbb{R},\leavevmode\nobreak\ \mathbb{P}$-a.s.

Given this setup we next define our concept of equilibrium in genre of [22].

######  Definition 4.2 (MF-CRRA-Forward relative performance equilibrium (for
the generic manager)).

Let $(\overline{X}_{t})_{t\geqslant 0}$ and
$(\overline{\Gamma}_{t})_{t\geqslant 0}$ be the $\mathbb{F}^{B}$-adapted
square integrable stochastic processes representing the geometric average
wealth and geometric average consumption of the continuum of agents
respectively. Let $(\pi,c)\in\mathcal{A}^{\mathrm{MF}}$ and $X^{\pi,c}$ solve
(4.2) with $\pi$ and $c$.

Set a $\mathbb{F}^{\mathrm{MF}}$-progressively measurable random field
$(Q(x,t))_{t\geqslant 0}$, satisfying the dynamics

$\displaystyle Q(x,t)$ $\displaystyle=U(x,t)+\int_{0}^{t}V(\hat{c}_{s}x,s)ds,$
(4.3)

where $\hat{c}_{t}=c_{t}\overline{\Gamma_{t}}^{-\theta}$ and
$U,V:\Omega\times\mathbb{R}\times[0,\infty)\to\mathbb{R}$ are two other
$\mathbb{F}$-progressively measurable random fields.

The field $Q$ is an _MF-forward relative performance_ for the _generic
manager_ if, for all $t\geqslant 0$, the following conditions hold:

  1. i)

The mappings $x\mapsto U(x,t),y\mapsto V(y,t)$, are $\mathbb{P}$-a.s. strictly
increasing and strictly concave;

  2. ii)

For any $(\pi,c)\in\mathcal{A}^{\textrm{MF}}$,
$Q(X_{t}^{\pi,c}\overline{X}_{t}^{\theta},t)$ is a (local) supermartingale and
$X^{\pi,c}$ is the _generic agent’s_ wealth process solving (4.2) for the
strategy $(\pi,c)$;

  3. iii)

There exists $(\pi^{*},c^{*})\in\mathcal{A}^{\textrm{MF}}$ such that
$Q(X^{*}_{t}\overline{X}_{t}^{\theta},t)$ is a (local) martingale where
$X^{*}$ solves (4.2) with $(\pi^{*},c^{*})$ plugged in as the strategy;

  4. iv)

We call $(\pi^{*},c^{*})$ of point iii) a _MF-equilibrium_ if
$\overline{X}_{t}=\exp\mathbb{E}[\log X^{*}_{t}|\mathcal{F}^{B}_{t}]$ and
$\overline{\Gamma}_{t}=\exp\mathbb{E}[\log c^{*}_{t}|\mathcal{F}^{B}_{t}]$ for
all $t\geqslant 0$, $\mathbb{P}$-a.s. where $X^{*}$ solves (4.2) with
$(\pi^{*},c^{*})$ plugged in as the strategy.

We denote the sextuple $(U,V,\pi^{*},c^{*},\overline{X},\overline{\Gamma})$
satisfying i)-iv) the _MF-Forward relative performance equilibrium_. The MF-
equilibrium is said to be _strong_ if $\pi^{*}$ is constant in time and
$\mathcal{F}^{0}$-measurable and $c^{*}_{t}$ is continuous and
$\mathcal{F}^{0}$-measurable for any $t\geqslant 0$.

The last point can be understood as a fixed point argument which creates a
compatibility condition between the generic agent within the continuum of
agents. In fact, conditionally on the BM $B$ each agent faces an independent
noise $W$ and an independent type vector $\zeta$. As argued in [47, 46, 22],
conditionally on $B$, all agents faces i.i.d. copies of the same optimisation
problem. The law of large numbers suggests that the geometric average terminal
wealth of the whole population should be $\exp\mathbb{E}[\log
X_{t}^{*}|\mathcal{F}^{B}_{t}]$.

Our construction allows us to identify $\exp\mathbb{E}[\log
X_{t}^{*}|\mathcal{F}^{B}_{t}]$ with a certain dynamics and, in turn, treat
this component as an additional uncontrolled state process. This avoids
altogether the conceptualisation of the master equation for models with
different types of agents. The latter is left for future research.

It is unclear if the definition for forward MFG we propose is sufficiently
tractable (concretely point iv)) to address the case of random coefficients or
the general setting of a forward utility map with a full Itô-field
representation (with volatility). This same issue is already present in the
upstream works [47, 46] if one wanted to generalise to random coefficients
there. Nonetheless, the definition works very well within the Merton market
model setup, yielding a particularly tractable framework of study to
understand the inherent difficulties of the general case. Additionally, our
goal is to further interpret the results as this is one of the 1st works
combining MFGs with forward performance criteria (also [22]).

###  4.3 Solving the optimisation problem

We now present the main result of this section which is the existence of a
_MF-Forward relative performance equilibrium_ for the generic manager
according to Definition 4.2 within the context of a structural assumption
similar to that of Equation (3.8).

From the methodological point of view, the problem is solved as before. Apply
Itô’s formula to $Q(Z^{\pi,c}_{t},t)$, determine the optimal strategy
$(\pi^{*},c^{*})$ and the consistency condition (the PDE) for $Q$ such that
the first three conditions of Definition 4.2 hold. The last condition, to show
that $(\pi^{*},c^{*})$ is indeed the MFG Forward equilibrium will follow by
construction as the whole formulation is built with the fixed-point condition
embedded from the start.

We next present the separable power factor form assumption for $U$ and $V$.

######  Assumption 4.3.

Let $U(x,t),V(t,x)$ admit the following form, for $x,y\in\mathbb{R}$, $t\geq
0$,

$\displaystyle
U(x,t)=\frac{1}{1-\frac{1}{\delta}}x^{1-\frac{1}{\delta}}f(t),\qquad
V(y,t)=\frac{1}{\epsilon}\frac{1}{1-\frac{1}{\delta}}y^{1-\frac{1}{\delta}}g(t),$
(4.4)

for $t\mapsto f(t)$ differentiable for any $t\geqslant 0$ and $f(0)=1$, for
some random variable $\epsilon>0$ and $g(t)=f(t)^{\kappa}$ for some
$\kappa\in\mathbb{R}$ (where we have $g(0)=1^{\kappa}=1$).

The parameter $\delta>0,\delta\neq 1$ represents the agent’s risk tolerance
while $\epsilon>0$, associated to the consumption element, represents the
relative importance the agent assigns to the consumption compared to wealth.
The parameter $\kappa$ represent the market consumption intensity.

In view of the previous section, this assumption is a natural one to make in
order to solve (4.4) below (see additionally [26], [4, Definition 2.9] or [16,
Section 4]).

######  Theorem 4.4.

Assume that $\delta>0,\leavevmode\nobreak\ \theta\in[0,1],\leavevmode\nobreak\
\mu>0,\leavevmode\nobreak\ \sigma\geqslant 0,\leavevmode\nobreak\ \nu\geqslant
0$ such that $\sigma^{2}+\nu^{2}>0$. Define constants
$\psi^{\sigma},\varphi^{\sigma},\psi^{\mu},\varphi^{\mu}$, assume they are
finite and have explicit form given by

$\displaystyle\psi^{\sigma}$
$\displaystyle:=\mathbb{E}\Big{[}\theta(1-\delta)\frac{\sigma^{2}}{\nu^{2}+\sigma^{2}}\Big{]},\quad\varphi^{\sigma}:=\mathbb{E}\Big{[}\delta\frac{\mu\sigma}{\nu^{2}+\sigma^{2}}\Big{]},\quad$
$\displaystyle\psi^{\mu}$
$\displaystyle:=\mathbb{E}\Big{[}\theta(1-\delta)\frac{\mu\sigma}{\nu^{2}+\sigma^{2}}\Big{]},\quad\varphi^{\mu}:=\mathbb{E}\Big{[}\delta\frac{\mu^{2}}{\nu^{2}+\sigma^{2}}\Big{]}.$

Assume further $\psi^{\sigma}\neq 1$ and define for some
$\kappa\in\mathbb{R}$,

$\displaystyle\pi^{*}$
$\displaystyle=\frac{1}{{\nu^{2}+\sigma^{2}}}\Big{(}\theta(1-\delta)\sigma\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}+\mu\delta\Big{)},$
(4.5) $\displaystyle c^{*}_{t}$
$\displaystyle=\begin{cases}\bigg{(}\frac{1}{\beta}+\Big{(}\frac{1}{\lambda}-\frac{1}{\beta}\Big{)}e^{-(\kappa-1)\beta
t}\bigg{)}^{-1},&\beta\neq 0,\\\
\big{(}(\kappa-1)t+\frac{1}{\lambda}\big{)}^{-1},&\beta=0.\end{cases}$ (4.6)

where

$\displaystyle\lambda$
$\displaystyle=\epsilon^{-\delta}\big{(}e^{\mathbb{E}[\delta\log\epsilon]}\big{)}^{\frac{\theta(1-\delta)}{\mathbb{E}[\theta(1-\delta)]-1}},\qquad\beta=\frac{\theta(1-\delta)}{\mathbb{E}[\theta(1-\delta)]-1}\mathbb{E}[\rho\delta]-\rho\delta,$

and

$\displaystyle\rho$
$\displaystyle=-\Big{(}1-\frac{1}{\delta}\Big{)}\bigg{(}\theta\overline{\mu\pi^{*}}-\frac{\theta}{2}\overline{\Sigma(\pi^{*})^{2}}-\frac{\theta\sigma\overline{\sigma\pi^{*}}(\mu-\theta\sigma\overline{\sigma\pi^{*}})}{\nu^{2}+\sigma^{2}}-\frac{\theta^{2}}{2}\big{(}\overline{\sigma\pi^{*}}\big{)}^{2}$
(4.7)
$\displaystyle\qquad\qquad-\delta\frac{(\mu-\theta\sigma\overline{\sigma\pi^{*}})^{2}}{2(\nu^{2}+\sigma^{2})}-\frac{1}{2\delta}\Big{(}\theta\overline{\sigma\pi^{*}}\Big{)}^{2}\Big{(}\frac{\sigma^{2}}{\nu^{2}+\sigma^{2}}-1\Big{)}\bigg{)},$

where $\overline{\sigma\pi^{*}},\leavevmode\nobreak\ \overline{\mu\pi^{*}}$
and $\overline{\Sigma(\pi^{*})^{2}}$ are given by (4.12), (4.20) and (4.21)
respectively.

Take the field $Q$ of (4.3) with $U,V$ satisfying Assumptions 4.1 and 4.3 (for
the $\kappa$ above), and the consistency PDE

$\displaystyle U_{t}$
$\displaystyle=\theta\bigg{(}\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}\psi^{\mu}+\varphi^{\mu}+\frac{\sigma\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}(\mu-\theta\sigma\frac{\varphi^{\sigma}}{1-\psi^{\sigma}})}{\nu^{2}+\sigma^{2}}-\frac{\theta}{2}\overline{\Sigma(\pi^{*})^{2}}-\frac{\theta}{2}\Big{(}\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}\Big{)}^{2}\bigg{)}xU_{x}$
$\displaystyle\qquad\qquad+\frac{\big{(}\mu-\theta\sigma\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}\big{)}^{2}}{2(\nu^{2}+\sigma^{2})}\frac{(U_{x})^{2}}{U_{xx}}+\frac{1}{2}\cdot\theta^{2}\Big{(}\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}\Big{)}^{2}\Big{(}\frac{\sigma^{2}}{\nu^{2}+\sigma^{2}}-1\Big{)}x^{2}U_{xx}$
(4.8) $\displaystyle\qquad\qquad+\theta
U_{x}^{i}\bar{c}_{t}-\widetilde{V}(U_{x},t),$

where $\overline{\Sigma(\pi^{*})^{2}}$ is given by (4.21),
$\bar{c}_{t}=\mathbb{E}[c^{*}_{t}]$, where $c^{*}$ is from (4.6) and
$\widetilde{V}$ is Fenchel-Legendre transform of $V$ (with initial condition
of power type according to Assumption 4.3 with $t=0$).

Then, there exists a unique (parameterised by $\kappa$) strong MF-Forward CRRA
relative performance equilibrium in the sense of Definition 4.2. The MF-
equilibrium strategy is given by $(\pi^{*},c^{*})$ from (4.5), (4.6) and MF-
forward CRRA relative performance utility is given by $Q$, for $U,V$
satisfying (4.4) and
$\overline{X}_{t}=\exp{\mathbb{E}\big{[}\log(X_{t}^{\pi^{*},c^{*}})\big{]}},\leavevmode\nobreak\
\overline{\Gamma}_{t}=\exp{\mathbb{E}[\log(c_{t}^{*})]}$. Moreover, MF-
equilibrium strategy is strong, if either $\kappa>1$ or
$\kappa<1,\leavevmode\nobreak\ \beta>\lambda$.

If $\psi^{\sigma}=1$, then there exists no strong MF-equilibrium.

###### Proof.

We proceed in several steps in order to construct the constant MF-equilibrium.
To that end we must solve ii)-iii) in Definition 4.2 for given processes
$\overline{X},\overline{\Gamma}$ associated to some
$(\pi,c)\in\mathcal{A}_{\textrm{MF}}$. Condition iv), for MF-equilibrium
allows us to focus only on processes of the form
$\overline{X}_{t}=\exp\mathbb{E}[\log X_{t}|\mathcal{F}^{B}_{t}]$ and
$\overline{\Gamma}_{t}=\exp\mathbb{E}[\log c_{t}|\mathcal{F}^{B}_{t}]$ where
$X$ solves (4.2) for some strategy $(\pi,c)\in\mathcal{A}_{MF}$.

_Step 0. The dynamics of the average wealth process._ To solve the above
problem given $(\overline{X}_{t})_{t\geqslant 0}$ it suffices to restrict
ourselves to processes $(\overline{X}_{t})_{t\geqslant 0}$ satisfying
$\overline{X}_{t}=\exp\mathbb{E}[\log
X^{\pi}_{t}|\mathcal{F}^{B}_{t}],\leavevmode\nobreak\ \mathbb{P}$-a.s. We then
have by applying Itô’s formula and arguments from [47] $\mathbb{P}$-a.s.

$\displaystyle\overline{X}_{t}$ $\displaystyle=\exp\mathbb{E}[\log
X_{t}|\mathcal{F}^{B}_{t}]$
$\displaystyle=\exp\mathbb{E}\Big{[}\log\xi+\int_{0}^{t}\big{(}\mu\pi_{s}-\frac{1}{2}\pi_{s}^{2}(\nu^{2}+\sigma^{2})\big{)}ds+\int_{0}^{t}\nu\pi_{s}dW_{s}+\int_{0}^{t}\sigma\pi_{s}dB_{s}-\int_{0}^{t}c_{s}ds\Big{|}\mathcal{F}^{B}_{t}\Big{]}$
$\displaystyle=\exp\Big{[}\overline{\log\xi}+\int_{0}^{t}\big{(}\overline{\mu\pi}-\frac{1}{2}\overline{\Sigma\pi^{2}}\big{)}ds+\int_{0}^{t}\overline{\sigma\pi}dB_{s}-\int_{0}^{t}\overline{c}_{s}ds\Big{]}$
$\displaystyle=\overline{\xi}+\Big{(}\int_{0}^{t}\eta\overline{X}_{s}ds+\int_{0}^{t}\overline{\sigma\pi}\overline{X}_{s}dB_{s}-\int_{0}^{t}\bar{c}_{s}\overline{X}_{s}ds\Big{)},$

where, for consistency of notation with respect to the previous section, we
denote

$\displaystyle\eta$
$\displaystyle:=\overline{\mu\pi}-\frac{1}{2}(\overline{\Sigma\pi^{2}}-\overline{\sigma\pi}^{2}),\quad\overline{\xi}:=\exp\mathbb{E}[\log\xi],\quad\overline{\mu\pi}:=\mathbb{E}[\mu\pi],\quad\overline{\sigma\pi}:=\mathbb{E}[\sigma\pi],\quad\bar{c}=\mathbb{E}[c].$

Hence for $(\pi,c)\in\mathcal{A}^{\textrm{MF}}$ we can define the process
$Z^{\pi,c}=X^{\pi,c}\overline{X^{\pi,c}}^{-\theta}$ and by Itô formula we
derive its dynamics to obtain

$\displaystyle\frac{dZ^{\pi,c}_{t}}{Z^{\pi,c}_{t}}$
$\displaystyle=\big{(}\mu\pi_{t}-\theta\overline{\mu\pi}+\frac{\theta}{2}\overline{\Sigma\pi^{2}}+\frac{\theta^{2}}{2}\overline{\sigma\pi}^{2}-\theta\sigma\pi_{t}\overline{\sigma\pi}\big{)}dt+\nu\pi_{t}dW_{t}+\big{(}\sigma\pi_{t}-\theta\overline{\sigma\pi}\big{)}dB_{t},$
$\displaystyle\qquad-(c_{t}-\theta\bar{c}_{t})dt,\quad
Z^{\pi,c}_{0}=\xi(\overline{\xi})^{-\theta},$

and solve the MFG Forward performance problem in Definition 4.2 with its help.

Applying Itô’s formula to $U(Z^{\pi,c}_{t},t)$ yields

$\displaystyle dQ(Z^{\pi,c}_{t},t)$
$\displaystyle=U_{t}(Z^{\pi,c}_{t},t)dt+U_{x}(Z^{\pi,c}_{t},t)dZ^{\pi,c}_{t}+\frac{1}{2}U_{xx}(Z^{\pi,c}_{t},t)d\langle
Z^{\pi,c}_{t}\rangle+V(Z^{\pi,c}_{t},t)dt$
$\displaystyle=\Big{[}U_{t}(Z^{\pi,c}_{t},t)+U_{x}(Z^{\pi,c}_{t},t)\big{(}\mu\pi_{t}-\theta\overline{\mu\pi}+\frac{\theta}{2}\overline{\Sigma\pi^{2}}+\frac{\theta^{2}}{2}\overline{\sigma\pi}^{2}-\theta\sigma\pi_{t}\overline{\sigma\pi}\big{)}Z^{\pi,c}_{t}$
$\displaystyle\qquad+\frac{1}{2}U_{xx}(Z^{\pi,c}_{t},t)\Big{(}(\nu\pi_{t})^{2}+\big{(}\sigma\pi_{t}-\theta\overline{\sigma\pi}\big{)}^{2}\Big{)}(Z^{\pi,c}_{t})^{2}\Big{]}dt$
(4.9)
$\displaystyle\qquad+U_{x}(Z^{\pi,c}_{t},t)\nu\pi_{t}Z^{\pi,c}_{t}dW_{t}+U_{x}(Z^{\pi,c}_{t},t)\big{(}\sigma\pi_{t}-\theta\overline{\sigma\pi}\big{)}Z^{\pi,c}_{t}dB_{t}$
$\displaystyle\qquad+U_{x}(Z^{\pi,c}_{t},t)(c_{t}-\theta\bar{c}_{t})dt+V(\hat{c}_{t}Z^{\pi,c}_{t},t)dt,$

with
$U(Z^{\pi,c}_{0},0)=U(\xi\big{(}\bar{\xi})^{-\theta},0\big{)}=\frac{1}{1-\frac{1}{\delta}}\big{(}\xi(\bar{\xi})^{-\theta}\big{)}^{1-\frac{1}{\delta}}$
and we used that the $B,W$ are all i.i.d.

_Step 1. Finding the candidate best responses strategies $\pi^{*},c^{*}$._ As
before, the process $U(Z^{\pi,c}_{t},t)$ becomes a Martingale at the optimum
$\pi$. Direct computations using first order conditions
($\partial_{\pi}\textrm{``drift''}=\partial_{c}\textrm{``drift''}=0$) yield

$\displaystyle\begin{cases}0+U_{x}\cdot\big{(}\mu-0-\theta\sigma\overline{\sigma\pi}\big{)}Z^{\pi,c}_{t}+\frac{1}{2}U_{xx}\Big{(}2\pi\nu^{2}+2\big{(}\sigma\pi_{t}-\theta\overline{\sigma\pi}\big{)}\sigma\Big{)}\big{(}Z^{\pi,c}_{t}\big{)}^{2}=0,\\\
-U_{x}(Z^{\pi,c}_{t},t)Z^{\pi,c}_{t}+V_{x}(\hat{c}_{t}Z^{\pi,c}_{t},t)\frac{Z^{\pi,c}_{t}}{\big{(}\overline{c}^{(-i)}_{t}\big{)}^{\theta_{i}}}=0,\end{cases}$
$\displaystyle\Rightarrow$
$\displaystyle\qquad\begin{cases}\pi_{t}&=\frac{1}{\nu^{2}+\sigma^{2}}\Big{(}\theta\sigma\overline{\sigma\pi}+(\mu-\theta\sigma\overline{\sigma\pi})\dfrac{U_{x}}{U_{xx}Z^{\pi,c}_{t}}\Big{)},\\\
c_{t}&=\dfrac{(V_{x})^{-1}\Big{(}U_{x}(Z^{\pi,c}_{t},t)\tilde{c}_{t}^{\theta},t\Big{)}\tilde{c}_{t}^{\theta}}{Z^{\pi,c}_{t}}.\end{cases}$
(4.10)

Now we inject into the first equation the CRRA constraint
$U_{x}/U_{xx}=-\delta x$ for any $t\geqslant 0$ $\mathbb{P}$-a.s. and use the
Assumption 4.3 to get

$\displaystyle\pi_{t}=\frac{1}{\nu^{2}+\sigma^{2}}\Big{(}\theta\sigma\overline{\sigma\pi}+(\mu-\theta\sigma\overline{\sigma\pi})\delta\Big{)}\quad\textrm{and}\quad
c_{t}=\tilde{c}_{t}^{\theta(1-\delta)}{f(t)}^{\delta(\kappa-1)},$

By inspection it is clear that $\pi^{*}$ is a
$\mathcal{F}_{0}^{\textrm{MF}}$-measurable RV which is independent of time and
is well defined as long as $\overline{\sigma\pi}$ is finite. At the same time
the derivation of the closed form of the optimal consumption needs further
investigation.

_Step 2. The optimality of the strategy._ The argument is similar to that in
[22]. The original constant strategy $\pi$ is a MF-equilibrium if and only if
for all $t\geqslant 0,\leavevmode\nobreak\ \mathbb{P}$-a.s.

$\displaystyle\begin{cases}\mathbb{E}[\log
X^{\pi,c}_{t}|\mathcal{F}^{B}_{t}]=\mathbb{E}[\log
X^{\pi^{*},c^{*}}_{t}|\mathcal{F}^{B}_{t}],\\\ \mathbb{E}[\log
c_{t}|\mathcal{F}^{B}_{t}]=\mathbb{E}[\log
c^{*}_{t}|\mathcal{F}^{B}_{t}],\end{cases}$
$\displaystyle\Leftrightarrow\quad$
$\displaystyle\begin{cases}\overline{\log\xi}+\big{(}\overline{\mu\pi}-\frac{1}{2}\overline{\Sigma\pi^{2}}\big{)}t+\overline{\sigma\pi}B_{t}-\int_{0}^{t}\bar{c}_{s}ds=\overline{\log\xi}+\big{(}\overline{\mu\pi^{*}}-\frac{1}{2}\overline{\Sigma(\pi^{*})^{2}}\big{)}t+\overline{\sigma\pi^{*}}B_{t}-\int_{0}^{t}\overline{c^{*}_{s}}ds,\\\
\tilde{c}_{t}=\tilde{c}^{*}_{t},\end{cases}$

where we denote $\tilde{c}_{t}:=\exp\mathbb{E}[\log
c_{t}|\mathcal{F}^{B}_{t}]=\exp\mathbb{E}[\log
c^{*}_{t}|\mathcal{F}^{B}_{t}]=:\tilde{c}^{*}_{t}$. After taking expectations
in the first equation it follows that $\pi$ is a MG-equilibrium if and only if
the following three conditions hold $\mathbb{P}$-a.s.

$\displaystyle\begin{cases}\overline{\sigma\pi}=\overline{\sigma\pi^{*}},\big{(}\overline{\mu\pi}-\frac{1}{2}\overline{\Sigma\pi^{2}}\big{)}t-\int_{0}^{t}\overline{c_{s}}ds=\big{(}\overline{\mu\pi^{*}}-\frac{1}{2}\overline{\Sigma(\pi^{*})^{2}}\big{)}t-\int_{0}^{t}\overline{c^{*}_{s}}ds,\\\
\tilde{c}_{t}=\tilde{c}^{*}_{t}.\end{cases}$ (4.11)

Using (4.10) (with $U_{x}/U_{xx}=-\delta x$ replaced in) one derives (using
the expressions $\varphi^{\sigma},\psi^{\sigma}$)

$\displaystyle\sigma\pi^{*}$
$\displaystyle=\theta(1-\delta)\frac{\sigma^{2}}{\nu^{2}+\sigma^{2}}\overline{\sigma\pi}+\delta\frac{\mu\sigma}{\nu^{2}+\sigma^{2}}\qquad\Rightarrow\quad\overline{\sigma\pi^{*}}=\overline{\sigma\pi}\psi^{\sigma}+\varphi^{\sigma}.$

Using that $\overline{\sigma\pi}=\overline{\sigma\pi^{*}}$ yields solvability
if
$\psi^{\sigma}=\mathbb{E}\big{[}\theta(1-\delta)\frac{\sigma^{2}}{\nu^{2}+\sigma^{2}}\big{]}\neq
1$. Thus

$\displaystyle\overline{\sigma\pi^{*}}=\overline{\sigma\pi}=\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}=\textrm{Const.},$
(4.12)

and the (4.5) follows. To exploit the next condition we are to solve PDE (4.4)
under Assumption 4.3. Together with the optimal candidate consumption we have

$\displaystyle\begin{cases}c^{*}_{t}=\epsilon^{-\delta}(\tilde{c}_{t})^{\theta(1-\delta)}f_{i}(t)^{(\kappa-1)\delta},\\\
f^{\prime}(t)+\Big{(}\chi-\theta\big{(}1-\frac{1}{\delta}\big{)}\big{(}\overline{\mu\pi}-\frac{1}{2}\overline{\Sigma\pi^{2}}-\bar{c}_{t}\big{)}\Big{)}f(t)+\frac{\epsilon^{-\delta}}{\delta}(\tilde{c}_{t})^{\theta(1-\delta)}f(t)^{(\kappa-1)\delta+1}=0,\end{cases}$
(4.13)

where

$\displaystyle\chi$
$\displaystyle:=\Big{(}1-\frac{1}{\delta}\Big{)}\bigg{(}\frac{\theta\sigma\overline{\sigma\pi^{*}}(\mu-\theta\sigma\overline{\sigma\pi^{*}})}{\nu^{2}+\sigma^{2}}+\frac{\theta^{2}}{2}\big{(}\overline{\sigma\pi^{*}}\big{)}^{2}$
$\displaystyle\qquad\qquad+\delta\frac{(\mu-\theta\sigma\overline{\sigma\pi^{*}})^{2}}{2(\nu^{2}+\sigma^{2})}+\frac{1}{2\delta}\Big{(}\theta\overline{\sigma\pi^{*}}\Big{)}^{2}\Big{(}\frac{\sigma^{2}}{\nu^{2}+\sigma^{2}}-1\Big{)}\bigg{)}.$

After we plug the first equation of (4.13) into the second we solve for

$\displaystyle
f(t)=\exp\Big{(}-\int_{0}^{t}\Big{(}\chi-\theta\big{(}1-\frac{1}{\delta}\big{)}\big{(}\overline{\mu\pi}-\frac{1}{2}\overline{\Sigma\pi^{2}}-\bar{c}_{s}\big{)}+\frac{1}{\delta}c^{*}_{s}\Big{)}ds\Big{)}.$

Now we substitute it back into the first equation of (4.13) to get

$\displaystyle
c^{*}_{t}\exp\big{(}(\kappa-1)\int_{0}^{t}c_{s}^{*}ds\big{)}=\epsilon^{-\delta}(\tilde{c}_{t})^{\theta(1-\delta)}e^{-\chi\delta(\kappa-1)t}\exp\Big{(}-\theta(1-\delta)(\kappa-1)\int_{0}^{t}\big{(}\overline{\mu\pi}-\frac{1}{2}\overline{\Sigma\pi^{2}}-\bar{c}_{s}\big{)}ds\Big{)}.$
(4.14)

Now we substitute according to second equilibrium identity of (4.11) to obtain

$\displaystyle
c^{*}_{t}\exp\big{(}(\kappa-1)\int_{0}^{t}c_{s}^{*}ds\big{)}=\epsilon^{-\delta}(\tilde{c}_{t})^{\theta(1-\delta)}e^{-\rho\delta(\kappa-1)t}\exp\big{(}\theta(1-\delta)(\kappa-1)\int_{0}^{t}\overline{c_{s}^{*}}ds\big{)},$
(4.15)

where

$\displaystyle\rho=\chi+\theta\big{(}1-\frac{1}{\delta}\big{)}\bigg{(}\overline{\mu\pi^{*}}-\frac{1}{2}\overline{\Sigma(\pi^{*})^{2}}\bigg{)}.$

Now we consequently take the logarithm, expectation and exponent on both sides
of (4.15) to get

$\displaystyle\tilde{c}^{*}_{t}\exp\big{(}(\kappa-1)\int_{0}^{t}\overline{c_{s}^{*}}ds\big{)}=\big{(}\widetilde{\epsilon^{\delta}}\big{)}^{-1}e^{-(\kappa-1)\overline{\rho\delta}t}\Big{(}\tilde{c}_{t}(\exp\big{(}\kappa-1)\int_{0}^{t}\overline{c_{s}^{*}}ds\big{)}\Big{)}^{\overline{\theta(1-\delta)}},$

where $\widetilde{\epsilon^{\delta}}=\exp\mathbb{E}[\delta\log\epsilon]$,
$\overline{\rho\delta}=\mathbb{E}[\rho\delta]$ and
$\overline{\theta(1-\delta)}=\mathbb{E}[\theta(1-\delta)]$. Using the last
equilibrium identity, $\tilde{c}_{t}=\tilde{c}^{*}_{t}$, we rewrite the
expression

$\displaystyle\tilde{c}^{*}_{t}\exp\big{(}(\kappa-1)\int_{0}^{t}\overline{c_{s}^{*}}ds\big{)}=\widetilde{\epsilon^{\delta}}^{\frac{1}{\overline{\theta(1-\delta)}-1}}e^{\frac{\overline{\rho\delta}}{\overline{\theta(1-\delta)}-1}(\kappa-1)t}.$
(4.16)

Plugging it back into (4.15) we obtain

$\displaystyle
c^{*}_{t}\exp\big{(}(\kappa-1)\int_{0}^{t}c_{s}^{*}ds\big{)}=\lambda
e^{\beta(\kappa-1)t},$ (4.17)

where

$\displaystyle\lambda$
$\displaystyle=\epsilon^{-\delta}\big{(}e^{\mathbb{E}[\delta\log\epsilon]}\big{)}^{\frac{\theta(1-\delta)}{\mathbb{E}[\theta(1-\delta)]-1}},\qquad\beta=\frac{\theta(1-\delta)}{\mathbb{E}[\theta(1-\delta)]-1}\mathbb{E}[\rho\delta]-\rho\delta.$

The same arguments as in Theorem 3.11 yield

$\displaystyle
c^{*}_{t}=\begin{cases}\bigg{(}\frac{1}{\beta}+\Big{(}\frac{1}{\lambda}-\frac{1}{\beta}\Big{)}e^{-(\kappa-1)\beta
t}\bigg{)}^{-1},&\beta\neq 0,\\\
\big{(}(\kappa-1)t+\frac{1}{\lambda}\big{)}^{-1},&\beta=0.\end{cases}$ (4.18)

As in Theorem 3.11, one finds that for certain combinations of parameters,
namely $\kappa<1,\leavevmode\nobreak\ \beta<\lambda$ and
$\kappa<1,\leavevmode\nobreak\ \beta=0$, the optimal consumption $c$ in not
continuous and even can be negative (however the first case is admissible, due
to the exponential convergence of the integral); see Table 5.1 below for
summary.

Finally, from the first line of (4.13), we get that

$\displaystyle
f(t)=\Big{(}\big{(}c_{t}\big{)}^{\frac{1}{\delta}}\big{(}\tilde{c}_{t}\big{)}^{\theta(\frac{1}{\delta}-1)}\epsilon\Big{)}^{\frac{1}{\kappa-1}}\quad\text{and
thus}\quad
g(t)=\Big{(}\big{(}c_{t}\big{)}^{\frac{1}{\delta}}\big{(}\tilde{c}_{t}\big{)}^{\theta(\frac{1}{\delta}-1)}\epsilon\Big{)}^{\frac{\kappa}{\kappa-1}}.$
(4.19)

Now we are left to conclude with the closed forms of $\overline{\mu\pi^{*}}$
and $\overline{\Sigma(\pi^{*})^{2}}$. First, we have

$\displaystyle\overline{\mu\pi^{*}}=\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}\psi^{\mu}+\varphi^{\mu}=\textrm{Const},$
(4.20)

using $\psi^{\mu},\varphi^{\mu}$. Finally we find the expression for
$\overline{\Sigma\pi^{2}}$. Multiplying separately (4.5) by $\sigma$ and
$\nu$, squaring, taking expectation and summing the results, we have

$\displaystyle\overline{\Sigma\pi^{2}}=\mathbb{E}\bigg{[}\frac{1}{\nu^{2}+\sigma^{2}}\Big{(}\theta(1-\delta)\sigma\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}+\mu\delta\Big{)}^{2}\bigg{]}.$
(4.21)

Now we find the explicit form of the expression $\rho$ by injecting these
identities in (4.7).

For the non-solvability statement, if the last equation in (4.20) has
$\psi^{\sigma}=1$ and $\varphi^{\sigma}\neq 0$ then the equation has no
solution and hence no constant MF-equilibrium exists. The case
$\psi^{\sigma}=1$ and $\varphi^{\sigma}=0$ is impossible. Since $\mu>0$ and
$\delta>0$ by assumption, it implies that $\sigma=0$ and hence that
$\psi^{\sigma}=0$ contradicting the condition $\psi^{\sigma}=1$.

_Step 3. The MFG forward performance process dynamics._ Injecting the
consistency PDE (4.4) in the expression for $dU(Z^{\pi,c}_{t},t)$ given in
(4.3) yields,

$\displaystyle dQ(Z^{\pi,c}_{t},t)$
$\displaystyle=U_{x}(Z^{\pi,c}_{t},t)\Big{(}\nu\pi_{t}dW_{t}+\big{(}\sigma\pi_{t}-\theta\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}\big{)}dB_{t}\Big{)}Z^{\pi,c}_{t}$
$\displaystyle\qquad\qquad+\frac{1}{2}U_{xx}(Z^{\pi,c}_{t},t)\frac{1}{(\nu^{2}+\sigma^{2})}\bigg{|}\pi_{t}(\nu^{2}+\sigma^{2})-\Big{(}\theta(1-\delta)\sigma\frac{\varphi^{\sigma}}{1-\psi^{\sigma}}+\mu\delta\Big{)}\bigg{|}^{2}(Z^{\pi,c}_{t})^{2}dt$
$\displaystyle\qquad\qquad+V(\hat{c}_{t}Z^{\pi,c}_{t},t)-U_{x}(Z^{\pi,c}_{t},t)c_{t}Z^{\pi,c}_{t}-\widetilde{V}(U_{x}(Z^{\pi,c}_{t},t),t).$

∎

We comment further on the admissibility of $c^{*}$ in Section 5. It happens
that $c^{*}$ can be admissible but not form a strong equilibria while
$\pi^{*}$ will always do so.

Now we are to get a simplification corollary for a single stock case.

######  Corollary 4.5 (Single stock).

Let $\mu,\sigma,\nu$ be deterministic with $\nu=0$ and $\mu,\sigma>0$. Then
$\lambda,\beta$ are expressed as

$\displaystyle\lambda$
$\displaystyle=\epsilon^{-\delta}\big{(}e^{\mathbb{E}[\delta\log\epsilon]}\big{)}^{\frac{\theta(1-\delta)}{\mathbb{E}[\theta(1-\delta)]-1}},\quad\beta=\frac{\mu^{2}}{2\sigma^{2}}(1-\delta)\big{(}1-\frac{\theta}{\theta_{\textrm{crit}}}\big{)}\Big{(}\delta+\frac{\theta}{\theta_{\textrm{crit}}}(1-\delta)\Big{)},$
(4.22)

where
$\theta_{\textrm{crit}}=\frac{1-\mathbb{E}[\theta(1-\delta)]}{\mathbb{E}[\delta]}$.

Then, if $\mathbb{E}[\theta(1-\delta)]\neq 1$ then a _MF-equilibrium_ exists,
with the optimal strategy $(\pi^{*},c^{*})$ given by

$\displaystyle\pi^{*}$
$\displaystyle=\frac{\mu}{\sigma^{2}}\Big{(}\delta+\frac{\theta}{\theta_{\textrm{crit}}}(1-\delta)\Big{)},$
(4.23) $\displaystyle c^{*}_{t}$
$\displaystyle=\begin{cases}\bigg{(}\frac{1}{\beta}+\Big{(}\frac{1}{\lambda}-\frac{1}{\beta}\Big{)}e^{-(\kappa-1)\beta
t}\bigg{)}^{-1},&\beta\neq 0,\\\
\big{(}(\kappa-1)t+\frac{1}{\lambda}\big{)}^{-1},&\beta=0.\end{cases}$ (4.24)

Furthermore, the MF-equilibrium is strong, if either $\kappa>1$, or
$\kappa<1,\leavevmode\nobreak\ \lambda<\beta$.

######  Remark 4.6 (On convergence of the Nash equilibria of $n$-player game).

We can see that the MF-equilibrium agrees with the limit of the $n$-player
game equilibrium strategies, as $n\to\infty,\leavevmode\nobreak\
\mathbb{P}$-a.s.. The respective parameters and functions of them converge to
the averages of type distributions by the strong law of large numbers.

##  5 A discussion of the results

In this section we focus in interpreting the results obtained by way of the
single stock case of the mean field game given by Corollary 4.5 (compare with
Corollary 3.13) and the parameters involved. On the one hand, it provides with
a valuable insight of the model, capturing the relations and their
interpretation, whilst one the other, avoids the complex intricate
dependencies of the general case. This case also allows for an easier
comparison with the results from [46] where standard utility maps were used.
For the sake of simplicity by writing $\pi,c$ we refer to $(\pi^{*},c^{*})$ \-
mean field single stock equilibria given by (4.23) and (4.24), if not
specified separately.

We will not concentrate on the complete analysis of optimal investment
strategy $\pi$ and omit some details on the optimal consumption strategy $c$
referring to the excellent discussions in the preceding works [46, 47, 22].
With this in mind we restrict our attention to the _consumption intensity_
$\kappa$ and its interplay with other parameters in scope of the optimal
consumption $c$. The case $\kappa=0$ is just that of [46, Section 4]and we
point the reader to the discussion there. The behaviours found here when
$\kappa\neq 0$ are not a symmetrised version of those of [46], as easily seen
in Table 5.1.

### Classical parameters

Recovering classical results. For the choice of $\theta=0$ (no performance
concerns and no competition) we recover the classical result by [51] and [59]
in (4.23)-(4.24): agents invest a constant fraction of wealth in the stock and
the consumption strategy is time-dependent but both investment and consumption
strategies are independent – this holds for the classic utility results in
[47]. Both here and in [22], the results differ to the mentioned ones as the
classical results encapsulate a dependence on the time horizon. When
$\kappa=0$ we recover the results in [46].

The market parameters $\beta,\lambda$. As in [46] and Corollary 4.5 here, we
can rewrite $\beta$ for

$\displaystyle\beta=\frac{\mu^{2}}{2\sigma^{2}}(1-\delta_{\text{eff}})\delta_{\text{eff}},\quad\textrm{where}\quad\delta_{\text{eff}}=(1-\frac{\theta}{\theta_{\text{crit}}})\delta+\frac{\theta}{\theta_{\text{crit}}}\quad\textrm{for}\quad\theta_{\textrm{crit}}=\frac{1-\mathbb{E}[\theta(1-\delta)]}{\mathbb{E}[\delta]}.$

We can see that this is the same form of $\beta$ that appears in the standard
Merton portfolio optimisation problem with no competition where
$\beta=\frac{\mu^{2}}{2\sigma^{2}}(1-\delta)\delta$. On the other hand, we can
rewrite $\beta$ as

$\displaystyle\beta=\frac{1}{2}\pi\mu(\delta-1)\Big{(}\frac{\theta}{\theta_{\textrm{crit}}}-1\Big{)},$

which shows that $\beta$ is in fact the expected return times extra risk and
an extra competition proportion that the agent is effectively willing to take
on top of the log risk-tolerant investor and
$\theta_{\text{crit}}$-competitive investor. We interpret $\beta$ as the
_expected effective portfolio return_. When it comes to $\lambda$, see (4.22),
the main relation to notice is its inverse dependence on the agent’s relative
perception of wealth relative to consumption $\epsilon$, scaled with the risk
tolerance $\delta$ and weighted by the perception of the population. We thus
characterise $\lambda$ as _effective relative perception of wealth with
respect to consumption_.

### The consumption intensity $\kappa$

Before we fully dive into analysis of $\kappa$, we differentiate between two
settings/interpretations. An agent in the absence of competition ($\theta=0$)
or purely looking for “best response” to the actions undertaken by the other
agents (see Remark 3.12) can determine its own idiosyncratic $\kappa$ while
for the Nash forward equilibrium it is required that all players interact
through a common $\kappa$. In either case we interpret $\kappa$ as a _market
consumption intensity_ observing that when the agent is free to choose
$\kappa$ for himself ($\theta=0$ case or just best response) the agent in fact
looks for the most suitable market investment-consumption environment via the
choice of $\kappa$.

#### The concept of Elasticity of intertemporal substitution (EIS)

We start by calculating one of the central parameters in models of dynamic
choice of consumption in macroeconomics and finance [62], the _elasticity of
intertemporal substitution_ (EIS) which measures the agent’s willingness to
substitute future consumption for present consumption in response to changes
in investment opportunities and is given by [38, 5]

$\displaystyle\text{EIS}:$
$\displaystyle=-\dfrac{d\big{(}\partial_{t}{c}_{t}/c_{t}\big{)}}{d\big{(}\partial_{t}V_{y}(c_{t}x,t)/V_{y}(c_{t}x,t)\big{)}},$

for some222The formula presented has stated the agent’s rate of consumption
equal to $c_{t}x$, where $c_{t}$ is a rate of consumption for unit of wealth
and $x>0$ some fixed amount of wealth. On the one hand, fixing the amount of
wealth helps us to purely investigate the relative effect of consumption, on
the other, it allows to assess its time differential to calculate EIS, in
contrast with $c_{t}X_{t}$ having an Itô-differential form and thus not
computable. $x>0$.

EIS measures the response of consumption growth to the deviation of _real
interest rate_ , captured by the denominator of the above expression. By
considering the dynamics of cost and worth of consumption in time, we
equivalently relate it to the borrowing and investing in present and future,
thus referring to the real interest rate. If $\text{EIS}<0$, the income of the
agent is inversely affected with respect to the consumption. As the real
interest rate goes up, the agent is to invest more, as he expects to get more
utility from savings/investment, hence decreasing his consumption. If the
return on savings/investment decreases, the agent feels that he can raise his
consumption level at the expense of wealth, this case is known as _income
effect_. On the contrary, $\text{EIS}>0$, the picture is reverse. Now the
increase in real interest rate encourages the agent to spend more, as he is
getting richer. On the contrary, if the savings are becoming less profitable,
the agent would prefer to decrease his consumption level, to avoid the
impoverishment. This in turn is known as _substitution effect_. Finally,
$\text{EIS}=0$ corresponds to a case of constant consumption rate where the
agent is indifferent about the real-time growth and would like to spend the
constant fraction of income all over the time.

In contrast with the classic utility optimisation approach, e.g., of [46]
where one has $\textrm{EIS}^{\textrm{classic}}=\delta$, the FPP approach
allows an intrinsic time-dependence of $V$, and thus

$\displaystyle\textrm{EIS}^{(\theta\neq
0)}_{t}=\frac{\textrm{EIS}^{(\theta=0)}}{1+\kappa\theta(1-\delta)\frac{1}{\gamma_{t}}},\quad\textrm{where}\quad\textrm{EIS}^{(\theta=0)}=(1-\kappa)\delta=(1-\kappa)\textrm{EIS}^{\textrm{classic}},$
(5.1)

where
$\gamma_{t}:=\dfrac{d\big{(}\partial_{t}{c}(t)/c(t)\big{)}}{d\big{(}\partial_{t}\tilde{c}(t)/\tilde{c}(t)\big{)}}$
denotes the _elasticity of conformity_ which is time dependent. Here
_elasticity of conformity_ captures how the change in agent’s consumption is
affected by the deviation in the consumption of the typical agent (as compared
against the geometric average $\tilde{c}=\exp\mathbb{E}[\log c_{t}]$.

The FPP here allows one to disentangle risk tolerance $\delta$ and
intertemporal elasticity of substitution (EIS); the standard standard utility
theory does not allow for this. This same feature has been reported in [37,
Slide 22] through the use of recursive utilities (in an more restricted Merton
market framework than ours but within MFGs) and which is close to our result
(5.1). When $\kappa=0$, the case of standard CRRA utility framework and
analysed in [46], immediately yields
$\text{EIS}=\delta=\textrm{EIS}^{\textrm{classic}}$.

#### The consumption intensity $\kappa$ under no performance concerns:
$\theta=0$ (no competition)

_Choosing different EIS and Risk Tolerance._ In the absence of competition,
i.e., when $\theta=0$, then (5.1) yields a non-standard form of
$\textrm{EIS}^{(\theta=0)}=(1-\kappa)\delta$, whereas standard utility theory
with CRRA type utilities yields $\textrm{EIS}^{\textrm{classic}}=\delta$. In
the no competition case, $\theta=0$, as we no longer require $\kappa$ to be
uniform and deterministic across the whole population, thus the agent can
choose in which consumption environment he is willing to invest, hence, as
opposed to [46], the agent can determine his own risk tolerance $\delta$ and
$\textrm{EIS}^{(\theta=0)}$. To our knowledge the only setting that allows
this is the recursive utility optimisation under Epstein-Zin preferences ([28,
29]), and forthcoming developments allowing for MFG competition, see [37, 36].

For $\kappa<1$ then $\textrm{EIS}^{(\theta=0)}>0$ and hence the agent is
subjected to _income_ effect, otherwise, when $\kappa>1$ then
$\textrm{EIS}^{(\theta=0)}<0$ and the agent allows for a _substitution_
effect. Finally, $\kappa=1$ reflects a constant consumption rate as
$\textrm{EIS}^{(\theta=0)}=0$. This last scenario is particularly important,
as the empirical research finds EIS in general to be around $0$ for the
average household (see [38], [13], [19]), reflecting an intuitive low risk
tolerance for the small investor.

#### The consumption intensity $\kappa$ under performance concerns:
$\theta\neq 0$

The agents are now synchronised on their consumption preference as they
compete ($\kappa$ is the same for all agents, see Remark 3.12). From (5.1) the
agent sees her consumption dynamics being re-scaled by competition
($\theta\neq 0$) and the market environment ($\kappa\neq 0)$. The agents can
access their EIS, (5.1), by adjusting the risk-competition preferences (the
elasticity of conformity $(\gamma_{t})_{t}$ depends on $\beta$ and $\lambda$
as chosen by the agents).

The parameter $\kappa$ as the "market consumption intensity". Appears in
Assumption 3.7 and 4.3 to influence the utility weight’s association via
$g(t)=f(t)^{\kappa}$. It reflects the weight in time the environment assigns
to the consumption process compared to the accumulation of wealth. Recalling
for simplicity the no-competition EIS,
$\textrm{EIS}^{(\theta=0)}=(1-\kappa)\delta$, we can see that $\kappa$ appears
as a linear factor scaling the allocation of consumption in time, justifying
its name.

When $\kappa>1$, the agent’s utility is more sensitive to the deviation of the
consumption than to wealth over time (as $g(t)=f(t)^{\kappa}$). At the same
time the agents are entitled to the effect of _substitution_. The market
environment a larger $\kappa$ (say $\kappa\geqslant 2$) encourages the agent
to accelerate its consumption policy, i.e., pushing the agent to reinforce
consumption in order to stabilise the level of consumption sooner (as $c$ is
asymptotic). The similar happens when $\kappa<1$, but the trend is reverse.
Here the market preferences are in line with wealth sensitivity: the evolution
of the wealth value is steeper than the one of consumption, and the agent
coordinates his decisions under the _income_ effect. For $\kappa\leqslant 0$
the agent is subjected to sharp changes in consumption dynamics, whilst
$0<\kappa<1$ encourages the steadier behaviour. Finally, $\kappa=1$ prescribes
the identical environment propensity for the wealth and consumption evolution
the agent is to follow.

In view of $c$, (4.24), $\kappa>2$ or $\kappa<0$ exacerbates the vanishing
role of the exponential term leading the agents to stabilise towards the
asymptotes ($\beta$ or zero) sooner than in the case $\kappa\in(0,2)$. See
Figure 5.1, where for different combinations of $\lambda,\beta$ where the
effect of $\kappa$ can be clearly seen; when $\kappa\in(0,2)$ the steady-state
of consumption (the asymptote) is reached much later. There is a noteworthy
macro-economics cause-effect phenomena here when having in mind the necessity
of central-planner mechanisms to stimulate consumption across the
economy/market.

The finer interplay of $\kappa$ with $\beta$ & $\lambda$ on the consumption
policy $c$.

Figure 5.1: Plots of optimal $c$ of (4.24) for various combinations of
parameters with a highlight on the monotonic behaviour of
$\kappa\mapsto(c_{t})_{t\geqslant 0}$ by way in which the asymptote is
reached.

We refer to [46, Section 4] that comprises the analysis for $\kappa=0$ which,
upon inspection, extends to our case when $\kappa<1$, thus we focus on the
case where $\kappa>1$, i.e., the market encourages the _substitution_ effect.
In terms of market behaviour, there are 4 cases to be distinguished:
$\beta>\lambda$, $0\leq\beta<\lambda$, $\beta<0<\lambda$ and $\beta=\lambda$
and we summarise them in Table 5.1 (by monotonic behaviour and associated
asymptotes, and admissibility). If $\beta=\lambda$ then $\kappa$ plays no role
in $c$, but it does affect the utility map. We emphasise that the expected
rate of return of the wealth ${\partial_{t}}\mathbb{E}[\log
X_{t}|\mathcal{F}_{0}]$ follows the opposite direction of the monotonicity of
$c$.

First, $c_{t}$ is increasing in time, when $\beta>\lambda$. Indeed, when the
relative perception of wealth is big, i.e., if $\epsilon\nearrow\infty$ then
$\lambda\searrow 0^{+}$, the agent’s consumption does not vanish and
monotonically increases to $\beta$; this is stark contrast with the case
$\beta>\lambda$ under $\kappa<1$, where $c_{t}\searrow 0^{+}$ monotonically as
$t\to\infty$ (compare Figure 5.1 b and d). From the beginning, the agent has a
focus on wealth, thus intends to consume less on the short-term (Figure 5.1
b). However, on the long-run the agent follows the market preference towards
consumption, and raises his proportion of consumption up to some
asymptotically stable level $(=\beta)$. As the agent’s wealth is increasing,
despite his preference towards wealth, he cannot resist the environment’s push
to spend more. This particular situation mimics a “keeping up with the
Joneses” [33] behaviour in a population of ultra-rich agents, where the
effective utility from getting richer comes from increasing level of
consumption and is accelerated by competition and the intensive consumption
environment.

The trend is similar when $\beta<\lambda$ for which the consumption is
monotonically decreasing. Indeed, when $\epsilon\searrow
0^{+},\leavevmode\nobreak\ \lambda\nearrow+\infty$, the agent prefers
consumption over wealth (while the market is pushing agents to consume
$\kappa>1$), so he starts to consume at a higher rate from the onset. However,
we distinguish two cases here. The agent with an expected effective portfolio
return $\beta\geqslant 0$, who seeks to stabilise his consumption and thus
decrease it asymptotically up to the safe rate $\beta$ (Figure 5.1 a). When
the agent chooses $\beta<0$, he monotonically reduces his rate of consumption
to $0$, even if he had started from a higher level. That is the case when the
agent is very risk-tolerant ($\delta>1$) or very competitive
($\theta>\theta_{\textrm{crit}}$) or both, so agent will accumulate his wealth
even if he is obtaining more utility-value from consumption over time (Figure
5.1 c).

Finally, when $\kappa=1$ or $\beta=\lambda,\leavevmode\nobreak\ c_{t}$ is
constant and equal to $\lambda$: the agent is indifferent towards the utility
from consumption, as EIS $=0$, the agent chooses to follow the relative
perception of consumption, captured by $\lambda$ through $\epsilon$ and his
risk and competition preferences.

Nonetheless, the agent’s utility is affected by $\kappa$. All cases are neatly
summarise in Table 5.1.

Parameters | Consumption $c$ of (4.24)
---|---
$\kappa$ | $\beta$ | $\lambda-\beta$ |  admissibility |  strong equilib. |  $c_{t}$ as $t\to\infty$
$>1$ | $>0$ | $>0$ | ✔ | ✔ | $\searrow\beta$
$<0$ | $\nearrow\beta$
$=0$ | constant throughout
$=0$ | $>0$ | $\searrow 0$
$<0$ | $>0$ | $\searrow 0$
$<1$ | $>0$ | $>0$ | ✔ | ✘ | finite-time blow-up
$<0$ | ✔ | $\searrow 0$
$=0$ | ✘ | constant throughout
$=0$ | $>0$ | ✘ | ✘ | finite-time blow-up
$<0$ | $>0$ | ✔ | ✔ | finite-time blow-up
$=1$ | any | any | ✔ | ✔ | constant throughout
Table 5.1: Summary of results for the behaviour of $(c_{t})_{t\geqslant 0}$ of
(4.24) as a map of its parameters. Here $\searrow,\nearrow$ denote the
decreasing/increasing monotonicity respectively and convergence from
above/below respectively (as $t\to\infty$). The optimal consumption may be
admissible but not satisfy the strong equilibrium condition (Definition 4.2).

#### The environment’s influence on the agents

One may wonder about the environment’s impact on the agent in the particular
case of EIS being close to zero. When the environment hardly distinguishes the
long-run preference for consumption from the one for wealth ($\kappa\to 1$),
the agent who wants to faster reach a constant level of consumption needs to
adapt by changing his risk and competition parameters in order to increase the
absolute value of $\beta$. For example, assume further a highly risk-tolerant
market with $\mathbb{E}[\delta]\to\infty$ and mutually independent types
$\theta$ and $\delta$, hence from (4.22) we have
$\theta_{\textrm{crit}}=\mathbb{E}[\theta]$. Then, the agent changes in the
following ways: the agent avoids the competition but accepts more risk
($\theta\to 0,\leavevmode\nobreak\ \delta>1$); or competes at the average
market level and accepts even more risk
($\theta\to\mathbb{E}[\theta],\leavevmode\nobreak\ \delta\gg 1$); or accepts
being highly competitive, but risk-averse
($\theta>\mathbb{E}[\theta],\leavevmode\nobreak\ \delta<1$).

Lastly, we note that $\beta$, as a quadratic function of
$\delta_{\text{eff}}$, is bounded from above by $\frac{\mu^{2}}{8\sigma^{2}}$
but is not bounded from below. That means that under a mild _substitution_
regime ($\kappa\to 1^{+}$), the rational agent cannot outperform the upper-
bound effective rate of portfolio return $\beta$. Nevertheless, by adjusting
the risk-competition parameters $(\delta,\theta)$, the agent can decrease his
consumption to zero as fast as he wants to.

At this point the market highly influences the choices of the player. If the
agent wants to maintain a desired level of consumption, he can do so by re-
setting his risk-competition preferences (and in accordance to remain optimal
“as a martingale”).

##  6 Open questions

From the construction we provided, here and in the much simpler case [22], it
is still open how to tackle the full MFG generalisation to market models
featuring random coefficients. Here the mean field aggregation approach
created in [47] is not possible anymore and a new tool is needed. We refer to
Remark 3.12 regarding the open issue of allowing agent to choose different
$\kappa$ parameters; if a solution is found for the finite-player game then
the fusing the multi-class approach of [58] to the types approach of [47, 46]
in regards of $\kappa$ could prove a tractable approach for the MFG case.

Generalising the dynamics of the forward performance utility map (3.1) or
(4.3) outside Assumption 4.1 to a fully Itô-dynamics and stochastic strategies
is also open. A crucial tool for such would be a general Itô-Wentzell-Lions
chain rule as developed in [21] and would likely build along [25] or [56]; or
alternatively, an approach similar to [16] can be taken where the volatility
of the FPP is exogenously postulated as the agent’s preferences (Forward-
Backward SDEs is the tool of choice there). It is also open exploring game
competition in the setting of [16] either for the finite-player game or the
MFG as in [34, 3, 22]. As pointed in [22, Section 5], many other questions can
be posed in this context of mean field FPP, ranging from possible non-
solvability [31], to risk-sharing pricing [10], indifference-pricing [48, 2,
15], ergodic problems [17] and associated numerics [35]. The cases and
analysis of [3] can be extrapolated to the MFG case as well.

A particular case of interest playing to the strengths of the forward
performance process framework, and which we do not explore here, is the
dynamic model selection problem. FPPs allow for a dynamic update of market
parameters to accommodate a switch in the market environment or the agent’s
perception of risk which the standard utility theory does not allow. In fact,
[22, Section 4.4] carries over to the construction we have provided here –
such would allow the agent to update the Market Consumption Intensity
parameter $\kappa$. An alternative view to this problem exploiting FPPs and
time dependent risk parameters can be found in [61].

Lastly, a comparative study between the capabilities of forward utility maps
versus Epstein-Zin preferences within the Krep-Porteus recursive utilities is
to the best of our knowledge an open question. As shown, the forward
performance criteria features the crucial property of the recursive utility
and hence this work stands to span new economics studies on EIS [1, 19, 62] .

## References

  * Aase [2016] K. K. Aase, “Recursive utility using the stochastic maximum principle,” _Quantitative Economics_ , vol. 7, no. 3, pp. 859–887, 2016.
  * Anthropelos [2014] M. Anthropelos, “Forward exponential performances: pricing and optimal risk sharing,” _SIAM J. Financial Math._ , vol. 5, no. 1, pp. 626–655, 2014. [Online]. Available: https://doi.org/10.1137/130910087
  * Anthropelos et al. [2020] M. Anthropelos, T. Geng, and T. Zariphopoulou, “Competition in fund management and forward relative performance criteria,” Nov 2020, researchGate preprint, DOI:10.13140/RG.2.2.20712.90889.
  * Avanesyan et al. [2020] L. Avanesyan, M. Shkolnikov, and R. Sircar, “Construction of a class of forward performance processes in stochastic factor models, and an extension of Widder’s theorem,” _Finance Stoch._ , vol. 24, no. 4, pp. 981–1011, 2020. [Online]. Available: https://doi.org/10.1007/s00780-020-00436-1
  * Azar [2018] S. A. Azar, “The Nexus Between the Elasticity of Intertemporal Substitution and the Coefficient of Relative Risk Aversion,” _International Journal of Financial Research_ , vol. 9, no. 3, pp. 98–102, July 2018. [Online]. Available: https://ideas.repec.org/a/jfr/ijfr11/v9y2018i3p98-102.html
  * Bank and Riedel [2001b] P. Bank and F. Riedel, “Existence and structure of stochastic equilibria with intertemporal substitution,” _Finance Stoch._ , vol. 5, no. 4, pp. 487–509, 2001. [Online]. Available: https://doi.org/10.1007/s007800100048
  * Bank and Riedel [2001a] ——, “Optimal consumption choice with intertemporal substitution,” _Ann. Appl. Probab._ , vol. 11, no. 3, pp. 750–788, 2001. [Online]. Available: https://doi.org/10.1214/aoap/1015345348
  * Benth et al. [2001] F. E. Benth, K. H. Karlsen, and K. Reikvam, “Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution,” _Finance Stoch._ , vol. 5, no. 4, pp. 447–467, 2001. [Online]. Available: https://doi.org/10.1007/s007800000032
  * Berrier and Tehranchi [2011] F. Berrier and M. Tehranchi, “Forward utility of investment and consumption,” _preprint_ , vol. 2, no. 9, p. 12, 2011.
  * Bielagk et al. [2017] J. Bielagk, A. Lionnet, and G. dos Reis, “Equilibrium pricing under relative performance concerns,” _SIAM J. Financial Math._ , vol. 8, no. 1, pp. 435–482, 2017. [Online]. Available: http://dx.doi.org/10.1137/16M1082536
  * Brennan [1975] M. J. Brennan, “The optimal number of securities in a risky asset portfolio when there are fixed costs of transacting: Theory and some empirical results,” _Journal of Financial and Quantitative analysis_ , pp. 483–496, 1975.
  * Campani and Garcia [2019] C. H. Campani and R. Garcia, “Approximate analytical solutions for consumption/investment problems under recursive utility and finite horizon,” _The North American Journal of Economics and Finance_ , vol. 48, pp. 364–384, 2019.
  * Campbell and Mankiw [1991] J. Y. Campbell and N. G. Mankiw, “The response of consumption to income: a cross-country investigation,” _European economic review_ , vol. 35, no. 4, pp. 723–756, 1991. [Online]. Available: http://www.sciencedirect.com/science/article/pii/001429219190033F
  * Carmona and Delarue [2018] R. Carmona and F. Delarue, _Probabilistic Theory of Mean Field Games with Applications I-II_. Springer, 2018.
  * Chong [2019] W. F. Chong, “Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences,” _Insurance: Mathematics and Economics_ , vol. 88, pp. 93–107, 2019.
  * Chong and Liang [2018] W. F. Chong and G. Liang, “Optimal investment and consumption with forward preferences and uncertain parameters,” _arXiv preprint arXiv:1807.01186_ , 2018.
  * Chong et al. [2019] W. F. Chong, Y. Hu, G. Liang, and T. Zariphopoulou, “An ergodic BSDE approach to forward entropic risk measures: representation and large-maturity behavior,” _Finance Stoch._ , vol. 23, no. 1, pp. 239–273, 2019. [Online]. Available: https://doi.org/10.1007/s00780-018-0377-3
  * Coval and Moskowitz [1999] J. D. Coval and T. J. Moskowitz, “Home bias at home: Local equity preference in domestic portfolios,” _The Journal of Finance_ , vol. 54, no. 6, pp. 2045–2073, 1999.
  * Cundy [2018] L. D. Cundy, “Essays on the elasticity of intertemporal substitution,” 2018, PhD dissertation. [Online]. Available: https://doi.org/10.17077/etd.2bg85sdu
  * Deng et al. [2020] C. Deng, X. Su, and C. Zhou, “Relative wealth concerns with partial information and heterogeneous priors,” _arXiv preprint arXiv:2007.11781_ , 2020.
  * dos Reis and Platonov [2019] G. dos Reis and V. Platonov, “Itô-Wentzell-Lions formula for measure dependent random fields under full and conditional measure flows,” _arXiv preprint arXiv:1910.01892_ , 2019.
  * dos Reis and Platonov [2020] ——, “Forward utilities and mean-field games under relative performance concerns,” _arXiv preprint arXiv:2005.09461_ , 2020.
  * Duffie and Epstein [1992a] D. Duffie and L. G. Epstein, “Asset pricing with stochastic differential utility,” _The Review of Financial Studies_ , vol. 5, no. 3, pp. 411–436, 1992.
  * Duffie and Epstein [1992b] ——, “Stochastic differential utility,” _Econometrica: Journal of the Econometric Society_ , pp. 353–394, 1992.
  * El Karoui and Mrad [2013] N. El Karoui and M. Mrad, “An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE,” _SIAM J. Financial Math._ , vol. 4, no. 1, pp. 697–736, 2013. [Online]. Available: https://doi.org/10.1137/10081143X
  * El Karoui et al. [2018] N. El Karoui, C. Hillairet, and M. Mrad, “Consistent utility of investment and consumption: a forward/backward SPDE viewpoint,” _Stochastics_ , vol. 90, no. 6, pp. 927–954, 2018. [Online]. Available: https://doi.org/10.1080/17442508.2018.1457676
  * El Karoui et al. [2019] ——, “Construction of an aggregate consistent utility, without Pareto optimality. Application to long-term yield curve modeling,” in _Frontiers in stochastic analysis—BSDEs, SPDEs and their applications_ , ser. Springer Proc. Math. Stat. Springer, Cham, 2019, vol. 289, pp. 169–199. [Online]. Available: https://doi.org/10.1007/978-3-030-22285-7_6
  * Epstein and Zin [1989] L. G. Epstein and S. E. Zin, “Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework,” _Econometrica_ , vol. 57, no. 4, pp. 937–969, 1989. [Online]. Available: http://www.jstor.org/stable/1913778
  * Epstein and Zin [1991] ——, “Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis,” _Journal of political Economy_ , vol. 99, no. 2, pp. 263–286, 1991.
  * Espinosa and Touzi [2015] G.-E. Espinosa and N. Touzi, “Optimal Investment under Relative Performance Concerns,” _Mathematical Finance_ , vol. 25, no. 2, pp. 221–257, 2015. [Online]. Available: http://dx.doi.org/10.1111/mafi.12034
  * Frei and dos Reis [2011] C. Frei and G. dos Reis, “A financial market with interacting investors: does an equilibrium exist?” _Mathematics and financial economics_ , vol. 4, no. 3, pp. 161–182, 2011. [Online]. Available: http://dx.doi.org/10.1007/s11579-011-0039-0
  * Fu et al. [2020] G. Fu, X. Su, and C. Zhou, “Mean field exponential utility game: A probabilistic approach,” _arXiv preprint arXiv:2006.07684_ , 2020.
  * Gali [1994] J. Gali, “Keeping up with the Joneses: Consumption externalities, portfolio choice, and asset prices,” _Journal of Money, Credit and Banking_ , pp. 1–8, 1994.
  * Geng and Zariphopoulou [2017] T. Geng and T. Zariphopoulou, “Passive and competitive investment strategies under relative forward performance criteria,” _Available at SSRN 2870040_ , 2017.
  * Gobet and Mrad [2018] E. Gobet and M. Mrad, “Convergence rate of strong approximations of compound random maps, application to SPDEs,” _Discrete Contin. Dyn. Syst. Ser. B_ , vol. 23, no. 10, pp. 4455–4476, 2018. [Online]. Available: https://doi.org/10.3934/dcdsb.2018171
  * Guéant [2021] O. Guéant, “Merton’s problem and comparison with peers: the expected utility framework and beyond,” 2021, forthcoming.
  * Guéant and Bismuth [2019] O. Guéant and A. Bismuth, “It’s all relative: mean field game extensions of merton’s problem,” Sep 2019, presentation at DEA Krakow, work forthcoming. [Online]. Available: https://www.oliviergueant.com/uploads/4/3/0/9/4309511/relative_merton_slides.pdf
  * Hall [1988] R. E. Hall, “Intertemporal substitution in consumption,” _Journal of Political Economy_ , vol. 96, no. 2, pp. 339–357, 1988. [Online]. Available: http://www.jstor.org/stable/1833112