# Dynamic capital allocation rules via BSDEs: an axiomatic approach

Elisa Mastrogiacomo111Dipartimento di Economia, Università dell’Insubria, via
Monte Generoso 71, 21100 Varese, Italy<EMAIL_ADDRESS>and
Emanuela Rosazza Gianin222Department of Statistics and Quantitative methods,
University of Milano Bicocca Via Bicocca degli Arcimboldi 8, 20126 Milano,
Italy<EMAIL_ADDRESS>
###### Abstract

In this paper, we study capital allocation for dynamic risk measures, with an
axiomatic approach but also by exploiting the relation between risk measures
and BSDEs. Although there is a wide literature on capital allocation rules in
a static setting and on dynamic risk measures, only a few recent papers on
capital allocation work in a dynamic setting and, moreover, those papers
mainly focus on the gradient approach. To fill this gap, we then discuss new
perspectives to the capital allocation problem going beyond those already
existing in the literature. In particular, we introduce and investigate a
general axiomatic approach to dynamic capital allocations as well as an
approach suitable for risk measures induced by $g$-expectations under weaker
assumptions than Gateaux differentiability.

Keywords: risk measures; capital allocation; BSDE; $g$-expectation;
subdifferential; gradient

##  1 Introduction

A relevant research stream related to risk measures is the capital allocation
problem, dealing with the problem of sharing in a suitable way the margin
required to hedge the riskiness of a position among the different sources of
riskiness of the aggregate position. For static risk measures the capital
allocation problem has been faced in an axiomatic way but also from an
empirical or a game-theoretical point of view (see [15, 19, 27, 41, 43, 10,
11] and the references therein). Although there is a wide literature on the
relation between dynamic risk measures and Backward Stochastic Differential
Equations (BSDEs, for short) and on capital allocation rules (CARs) in a
static setting, a systematic analysis of the dynamic setting has not yet been
extensively developed and only a few papers (see [6, 13, 29, 30, 32, 42]) on
capital allocation work in a dynamic setting. Even if the idea of using BSDEs
(or Backward Stochastic Volterra Integral Equations) for dynamic CARs can be
found already in [29, 30, 32], in these papers very specific CARs are
considered and the authors mainly investigate and cover the gradient approach.

Motivated by the relevance of dynamic risk measures and the need of a complete
study on CARs in a dynamic setting, in this paper we introduce an axiomatic
approach to dynamic capital allocation as well as an approach suitable for
risk measures induced by BSDES or, better, by $g$-expectations (see [34]). The
main goal of the paper is, therefore, to investigate from an axiomatic point
of view capital allocation rules of convex risk measures in a dynamic setting
by weakening the Gateaux differentiability condition of the risk measure and
going beyond the gradient approach. In particular, we generalize the axioms
introduced by [27] for capital allocations in a dynamic setting and
investigate their relation and compatibility with time-consistency of dynamic
risk measures. Furthermore, when the underlying risk measure is induced by a
BSDE governed by a non-smooth (but convex) driver, we introduce dynamic
capital allocation rules going beyond the gradient allocation (already
represented in terms of the driver in [29, 30, 32]).

The paper is organized as follows: in the next section we introduce notations
and review the main notions and results useful in the paper; in Section 3 we
introduce an axiomatic approach to capital allocation rules in a dynamic
setting and investigate existence of rules fulfilling some further axioms. In
Section 4, instead, we focus on dynamic capital allocation rules induced by
$g$-expectations and present some examples. Conclusions and final remarks are
provided in Section 5.

##  2 Notations and preliminaries

In this section we recall some basic notions and definitions that will be used
in the following sections. We first concern with the notion of dynamic risk
measures and the related properties and then move to basic facts about dynamic
measures and their connection to BSDEs.

Dynamic risk measures.

Let $T>0$ be a given future time horizon and let $(\Omega,\mathcal{F},P)$ be a
general probability space. Consider a continuous-time setting where time
evolves between $0$ and $T$ and let $(\mathcal{F}_{t})_{0\leq t\leq T}$ be a
filtration such that $\mathcal{F}_{0}=\\{\emptyset,\Omega\\}$ and
$\mathcal{F}_{T}=\mathcal{F}$. We will focus on risk measures quantifying the
riskiness of financial positions belonging to
$L^{\infty}(\mathcal{F}_{T})=L^{\infty}(\Omega,\mathcal{F}_{T},P)$, that is,
the space of essentially bounded random variables defined on
$(\Omega,\mathcal{F}_{T},P)$. All equalities and inequalities have to be
understood in the $P$-a.s. sense.

We recall that a static risk measure is a functional quantifying now the
riskiness of any position $X$ of maturity $T$, while a dynamic risk measure is
a functional quantifying the riskiness of $X$ at any time $t\in[0,T]$, taking
into account the whole information available up to time $t$. More precisely,
we recall the following definition by referring to [1, 4, 12, 16, 15, 17, 20,
23, 26], among many others, for a more detailed treatment on static and
dynamic convex risk measures.

###### Definition 1

A static risk measure is a functional
$\rho:L^{\infty}(\mathcal{F}_{T})\to\mathbb{R}$ satisfying some further
assumptions (e.g., monotonicity, cash-invariance, convexity).

A dynamic risk measure $(\rho_{t})_{t\in[0,T]}$ is a family of functionals

$\rho_{t}:L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t}),\quad
t\in[0,T],$

such that $\rho_{0}$ is a static risk measure and $\rho_{T}(X)=-X$ for any
$X\in L^{\infty}(\mathcal{F}_{T})$.

An incomplete list of desirable properties that are sometimes imposed to
dynamic risk measures $(\rho_{t})_{t\in[0,T]}$ are the following:

\- monotonicity: if $X,Y\in L^{\infty}(\mathcal{F}_{T})$ and $X\leq Y$, then
$\rho_{t}(X)\geq\rho_{t}(Y)$ for any $t\in[0,T]$.

\- cash-additivity: $\rho_{t}(X+m_{t})=\rho_{t}(X)-m_{t}$ for any $X\in
L^{\infty}(\mathcal{F}_{T})$, $m_{t}\in L^{\infty}(\mathcal{F}_{t})$ and
$t\in[0,T]$.

\- convexity: $\rho_{t}(\alpha
X+(1-\alpha)Y)\leq\alpha\rho_{t}(X)+(1-\alpha)\rho_{t}(Y)$ for any $X,Y\in
L^{\infty}(\mathcal{F}_{T})$, $t\in[0,T]$ and $\alpha\in[0,1]$.

\- normalization: $\rho_{t}(0)=0$, for any $t\in[0,T]$.

\- time-consistency: $\rho_{s}(X)=\rho_{s}(-\rho_{t}(X))$, for any $X\in
L^{\infty}(\mathcal{F}_{T})$, $0\leq s\leq t\leq T$.

\- weak time-consistency: $\rho_{s}(X)\leq\rho_{s}(-\rho_{t}(X))$, for any
$X\in L^{\infty}(\mathcal{F}_{T})$, $0\leq s\leq t\leq T$.

While monotonicity, cash-additivity, convexity and normalization are
essentially a translation to the dynamic setting of the corresponding static
axioms, time-consistency is peculiar to the dynamic setting and is a
recursivity property. Weak time-consistency, instead, is a weaker version of
recursivity.

In the following, with dynamic convex risk measures we mean any dynamic risk
measure satisfying monotonicity, cash-additivity, convexity, and
normalization. We recall the following results on the dual representations of
risk measures (see [5] and [20]).

Let $(\rho_{t})_{0\leq t\leq T}$ be a dynamic convex risk measure satisfying
continuity from below, that is, if $(X_{n})_{n\geq 0}\subseteq
L^{\infty}(\mathcal{F}_{T})$ is an increasing sequence with $X_{n}\nearrow X$,
then $\rho_{t}(X_{n})\to_{n\to+\infty}\rho_{t}(X)$, $P$-a.s. Then it can be
represented as

$\rho_{t}(X)=\operatorname*{ess\,max}_{Q\in\mathcal{Q}_{t}}\\{E_{Q}[-X|\mathcal{F}_{t}]-c_{t}(Q)\\},$
(1)

where $c_{t}$ is the so called (minimal) penalty term, defined as

$c_{t}(Q)\triangleq\operatorname*{ess\,sup}_{Y\in
L^{\infty}(\mathcal{F}_{T})}\\{E_{Q}[-Y|\mathcal{F}_{t}]-\rho_{t}(Y)\\},\quad
Q\in\mathcal{P}_{t},$ (2)

with $\mathcal{P}_{t}$ being the set of probability measures defined on
$(\Omega,\mathcal{F}_{T})$, absolutely continuous with respect to $P$ and such
that $Q|_{\mathcal{F}_{t}}=P$, and
$\mathcal{Q}_{t}\triangleq\\{Q\in\mathcal{P}_{t}:c_{t}(Q)\in
L^{\infty}(\mathcal{F}_{t})\\}$. Coherent risk measures correspond to penalty
terms belonging to $\\{0;+\infty\\}$ (see [17]).

BSDEs and their connection to dynamic risk measures.

Let us now recall the connection between dynamic risk measures and BSDEs. Let
$(B_{t})_{t\geq 0}$ be a standard $d$-dimensional Brownian motion defined on
the probability space $(\Omega,\mathcal{F},P)$. Denote by
$(\mathcal{F}_{t})_{t\geq 0}$ the natural Brownian filtration augmented by the
$P$-null sets. In the sequel, we identify a probability measure
$Q<\\!\\!\\!<P$ with its Radon-Nikodým density $\frac{{\rm d}Q}{{\rm d}P}$.
Since we are working in a Brownian setting, we also identify a probability
measure $Q$ equivalent to $P$ with the predictable process
$(q_{t})_{t\in[0,T]}$ induced by the stochastic exponential, i.e., such that

$\displaystyle
E\left[\left.\frac{dQ}{dP}\right|\mathcal{F}_{t}\right]=\mathcal{E}(q.B)_{t}=\exp\left(-\frac{1}{2}\int_{0}^{t}\|q_{s}\|^{2}ds+\int_{0}^{t}q_{s}dB_{s}\right).$

See Revuz and Yor [37] for details. Moreover, given a probability measure $Q$
on $(\Omega,\mathcal{F})$, we denote by $E_{Q}$ (respectively, $E_{Q}[\ \cdot\
|\mathcal{F}_{t}]$) the expected value (respectively, the conditional
expectation) with respect to $Q$. When the expectation is considered with
respect to the reference probability measure $P$, we will simply use the
notation $E$.

Pardoux and Peng [33] introduced equations of the following type (known as
BSDEs):

$\displaystyle
Y_{t}=X+\int_{t}^{T}g(s,Y_{s},Z_{s})ds-\int_{t}^{T}Z_{s}dB_{t},\quad 0\leq
t\leq T$ (3)

where $g:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{R}$
is generally called generator or driver, $T$ is the time horizon and $X\in
L^{2}(\mathcal{F}_{T})$ is a terminal condition, where
$L^{2}(\mathcal{F}_{T})$ is the space of all square integrable random
variables on $(\Omega,\mathcal{F}_{T},P)$. To simplify the notation we often
write $g(t,y,z)$ instead of $g(\omega,t,y,z)$.

We know from [33, Theorem 4.1] and [34, Theorem 35.1] or [22] that if $g$
satisfies the usual assumptions:

($g$1) $g$ Lipschitz continuous in $y$ and $z$, i.e., there exists a constant
$\mu>0$ such that $dP\times dt$, for any
$(y_{0},z_{0}),(y_{1},z_{1})\in\mathbb{R}\times\mathbb{R}^{d}$,

$|g(t,y_{0},z_{0})-g(t,y_{1},z_{1})|\leq\mu\left(|y_{0}-y_{1}|+\|z_{0}-z_{1}\|\right);$

($g$2) for any $(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$, $g(\cdot,y,z)$ is a
predictable process such that

$\displaystyle E\left[\int_{0}^{T}|g(s,y,z)|^{2}\,ds\right]<+\infty;$ (4)

($g$3) $g(t,y,0)=0$ for any $t\in[0,T]$,

then, for every $X\in L^{2}(\mathcal{F}_{T})$, the BSDE in (3) admits a unique
solution $(Y_{t}^{X},Z_{t}^{X})$ consisting of predictable processes with
values in $\mathbb{R}\times\mathbb{R}^{d}$ such that

$E\left[\int_{0}^{T}Y_{t}^{2}dt\right]<+\infty\quad\text{and}\quad
E\left[\int_{0}^{T}\|Z_{t}\|^{2}dt\right]<+\infty.$

Note that the existence and uniqueness of the solution is guaranteed also when
($g$3) is replaced by

($g$3’) $E\left[\int_{0}^{T}|g(s,y,0)|^{2}ds\right]<+\infty$ for any
$y\in\mathbb{R}$.

In such a case, the driver will be said to satisfy the non-normalized usual
assumptions.

Using Peng [34]’s terminology, the first component of the solution to the BSDE
(3)

$\mathcal{E}^{g}(X|\mathcal{F}_{t})\triangleq Y_{t}^{X}$

is called conditional $g$-expectation at time $t$. When $g(t,z)=\mu\|z\|$
(with $\mu>0$), $\mathcal{E}_{g}$ will be denoted by $\mathcal{E}_{\mu}$ (see,
e.g., [34]). Some relevant results about the link between $g$-conditional
expectations and dynamic risk measures can be found in [39, Section 4] and
[3].

As shown in [39], for a convex driver satisfying the usual assumptions
($g$1)-($g$3) and independence from $y$,

$\rho_{t}^{g}(X)\triangleq\mathcal{E}^{g}(-X|\mathcal{F}_{t}),\qquad\ X\in
L^{2}(\mathcal{F}_{T}),t\in[0,T],$

is a dynamic convex and time-consistent risk measure satisfying normalization.

##  3 Axiomatic approach to capital allocation: from a static to a dynamic
setting

In this section, we introduce and investigate an axiomatic approach to dynamic
capital allocation in the spirit of [27].

We start by recalling some theoretical aspects of capital allocation in a
static framework (see [15, 19, 11, 27, 41, 43], among others, for a detailed
treatment). Assume that a time horizon $T$ and a static risk measure $\rho$
have been fixed and let $X\in L^{\infty}(\mathcal{F}_{T})$ be a financial
position (with maturity $T$) which is formed by the sub-units (or business
lines or sub-portfolios) $X_{1},\dots,X_{n}\in L^{\infty}(\mathcal{F}_{T})$,
i.e. $X=\sum_{i=1}^{n}X_{i}$. For instance, we can think at $X$ as the profit
and loss of a portfolio composed by different assets and at $X_{1},...,X_{n}$
as at the profits and losses of the different assets, or at $X$ as the
riskiness of a firm and at $X_{1},...,X_{n}$ as at the different branches or
business lines of the firm.

The capital allocation problem consists then in finding a “suitable way” to
share the margin $\rho(X)$ among the different sub-units $X_{1},\dots,X_{n}$.
More concretely, it consists in finding $k_{1},\dots,k_{n}\in\mathbb{R}$ such
that $\rho(X)=k_{1}+\dots+k_{n}$ where each $k_{i}$ denotes the capital to be
allocated to $X_{i}$ or, in other words, the risk contribution of $X_{i}$ to
the total risk capital $\rho(X)$ of $X$.

More in general, given a convex risk measure $\rho$, Kalkbrener [27] defined
capital allocation rule (CAR) with respect to $\rho$ any functional
$\Lambda:L^{\infty}(\mathcal{F}_{T})\times
L^{\infty}(\mathcal{F}_{T})\to\mathbb{R}$ satisfying $\Lambda(X;X)=\rho(X)$.
In the terminology above, then, $\Lambda(X_{i};X)$ represents the capital to
be allocated to the sub-unit $X_{i}$ of the whole position $X$. However,
$\Lambda$ is defined for any pair $(X,Y)\in L^{\infty}(\mathcal{F}_{T})\times
L^{\infty}(\mathcal{F}_{T})$ where $X$ can be seen as a sub-portfolio333To be
more precise, given a position $Y$, $X$ is called sub-portfolio of $Y$ if
$Y=X+Z$ for some $Z$. In particular, in $L^{\infty}(\mathcal{F}_{T})$ any pair
$(X,Y)$ can be seen as sub-portfolio and portfolio, respectively. See [11]. of
the whole portfolio $Y$, and $\Lambda(X;Y)$ can be interpreted as the capital
allocated to the sub-portfolio $X$ to cover the riskiness of the global
portfolio $Y$. Furthermore, the condition $\Lambda(X;X)=\rho(X)$ imposed to a
CAR guarantees that for a stand-alone portfolio the capital to be allocated
corresponds exactly to the margin (or capital requirement) $\rho(X)$. Note
that a general CAR may fail to satisfy the requirement
$\rho(X)=\sum_{i=1}^{n}\Lambda(X_{i};X)$. If such a condition is fulfilled,
then the CAR is said to satisfy full allocation.

Before considering the dynamic version of capital allocation rules, we recall
two well-known and quite popular (static) CARs used in the literature: the
gradient and the Aumann-Shapley CARs (see [15, 19, 11, 27]). If $\rho$ is
Gateaux differentiable at $Y$, then:

  * •

the gradient CAR is defined as the directional derivative of $\rho$ at $Y$ in
the direction of $X$, i.e.

$\Lambda^{grad}(X;Y)\triangleq\lim_{h\to 0}\frac{\rho(Y+hX)-\rho(Y)}{h},$

and can be interpreted as the marginal contribution of sub-unit $X$ to the
overall risk $Y$;

  * •

the Aumann-Shapley CAR is defined as

$\Lambda^{AS}(X;Y)\triangleq\int_{0}^{1}\Lambda^{grad}(X;\gamma Y)d\gamma$

and somehow corresponds to the Aumann-Shapley value of game theory. We recall,
indeed, from Denault [19] that the Aumann-Shapley value for player/portfolio
$i$ is defined as

$\Lambda_{i}^{AS}\triangleq\int_{0}^{1}\frac{\partial
r}{\partial\delta_{i}}(\gamma\Delta)\,d\gamma,$

where $\delta\in\mathbb{R}^{n}_{+}$ represents the level of participation of
the $n$ players in a coalition, the components of
$\Delta\in\mathbb{R}^{n}_{+}$ the full participation of the players, and
$r:\mathbb{R}^{n}\to\mathbb{R}$ denotes a cost function. In the context of
fractional players, the Aumann-Shapley value can be then interpreted as the
average of the marginal costs of player/portfolio $i$, where the average is
taken with respect to $\gamma$, acting on the size of portfolio (from $0$ to
$\Delta$).

For non-Gateaux differentiable risk measures, subdifferential versions of the
previous (static) CARs are defined and studied in [11].

Although there is a wide literature on dynamic risk measures and on capital
allocation rules in a static setting, a systematic analysis of dynamic capital
allocations has not yet been extensively developed and only a few papers (see
[6, 13, 29, 30, 32, 42]) on capital allocation work in a dynamic setting.
Furthermore, in these papers very specific CARs are considered and the authors
mainly investigate and cover the gradient approach. For this reason, we start
providing an axiomatic approach to CARs in the dynamic setting in full
generality. In the next section, inspired by [29, 30, 32], we then introduce
an approach suitable for risk measures induced by BSDEs.

The general notion of a static capital allocation rule can be immediately
generalized to a dynamic framework.

###### Definition 2

Given a dynamic risk measure $(\rho_{t})_{t\in[0,T]}$, we define dynamic
capital allocation rule (CAR) associated to $(\rho_{t})_{t\in[0,T]}$ a family
$(\Lambda_{t})_{t\in[0,T]}$ of functionals

$\Lambda_{t}:L^{\infty}(\mathcal{F}_{T})\times L^{\infty}(\mathcal{F}_{T})\to
L^{\infty}(\mathcal{F}_{t})$

satisfying $\Lambda_{t}(X;X)=\rho_{t}(X)$ for any $X\in
L^{\infty}(\mathcal{F}_{T})$ and $t\in[0,T]$.

An audacious dynamic CAR, instead, will only satisfy
$\Lambda_{t}(X;X)\leq\rho_{t}(X)$ for any $X\in L^{\infty}(\mathcal{F}_{T})$
and $t\in[0,T]$ (see [11] for the static version).

Similarly to the static case, $\Lambda_{t}(X;Y)$ can be interpreted as the
(random) amount to be allocated to $X$ as a sub-portfolio of $Y$ at time $t$.
Differently from a static CAR where $\Lambda(X;Y)$ is deterministic, for a
dynamic CAR $\Lambda_{t}(X;Y)$ is a $\mathcal{F}_{t}$-measurable random
variable, hence taking into account all the information available up to time
$t$.

Here below we provide a list of desirable axioms which extend to the dynamic
setting those defined for static CARs in [27, 19, 11, 43]. In a static
setting, no-undercut and full allocation were already studied in [27] while
the others have been introduced and discussed later on.

\- monotonicity: if $X\leq Z$ then $\Lambda_{t}(X;Y)\geq\Lambda_{t}(Z;Y)$ for
any $Y\in L^{\infty}(\mathcal{F}_{T})$ and $t\in[0,T]$.

\- no-undercut: $\Lambda_{t}(X;Y)\leq\rho_{t}(X)$ for any $X,Y\in
L^{\infty}(\mathcal{F}_{T})$ and $t\in[0,T]$.

\- $\mathcal{F}_{t}$-riskless: $\Lambda_{t}(c_{t};Y)=-c_{t}$ for any $c_{t}\in
L^{\infty}(\mathcal{F}_{t}),Y\in L^{\infty}(\mathcal{F}_{T})$ and $t\in[0,T]$.

\- $\mathcal{F}_{t}$-1-cash-additivity:
$\Lambda_{t}(X+c_{t};Y)=\Lambda_{t}(X;Y)-c_{t}$ for any $c_{t}\in
L^{\infty}(\mathcal{F}_{t}),X,Y\in L^{\infty}(\mathcal{F}_{T})$ and
$t\in[0,T]$.

\- $\mathcal{F}_{t}$-cash-additivity:
$\Lambda_{t}(X+c_{t};Y+c_{t})=\Lambda_{t}(X;Y)+c_{t}$ for any $c_{t}\in
L^{\infty}(\mathcal{F}_{t}),X,Y\in L^{\infty}(\mathcal{F}_{T})$ and
$t\in[0,T]$.

\- full allocation:
$\Lambda_{t}(\sum_{i=1}^{n}Y_{i};Y)=\sum_{i=1}^{n}\Lambda_{t}(Y_{i};Y)$ for
any $Y_{1},...,Y_{n},Y\in L^{\infty}(\mathcal{F}_{T})$ such that
$Y=\sum_{i=1}^{n}Y_{i}$, and $t\in[0,T]$.

\- sub-allocation:
$\Lambda_{t}(\sum_{i=1}^{n}Y_{i};Y)\geq\sum_{i=1}^{n}\Lambda_{t}(Y_{i};Y)$ for
any $Y_{1},...,Y_{n},Y\in L^{\infty}(\mathcal{F}_{T})$ such that
$Y=\sum_{i=1}^{n}Y_{i}$, and $t\in[0,T]$.

\- weak convexity:
$\Lambda_{t}(\sum_{i=1}^{n}\alpha_{i}Y_{i};Y)\leq\sum_{i=1}^{n}\alpha_{i}\Lambda_{t}(Y_{i};Y)$
for any $\alpha_{i}\in[0,1]$ ($i=1,...,n$), $Y_{1},...,Y_{n},Y\in
L^{\infty}(\mathcal{F}_{T})$ satisfying $Y=\sum_{i=1}^{n}\alpha_{i}Y_{i}$ and
$\sum_{i=1}^{n}\alpha_{i}=1$, and $t\in[0,T]$.

Monotonicity means that the capital allocated to a position $Z$ has to be
smaller than the capital allocated to a riskier position $X$. No-undercut
translates the idea that, at any time $t$, the capital allocated to any $X$
considered as a sub-portfolio of $Y$ does not exceed the capital allocated to
$X$ considered as a stand-alone portfolio. In the terminology of Tsanakas
[43], such property guarantees that there is no incentive to split $X$ from
$Y$ because the capital requirement due to $X$ as a stand-alone portfolio
would be higher than the capital to be allocated to $X$ as a sub-portfolio of
$T$. $\mathcal{F}_{t}$-riskless means that the capital allocated to a, roughly
speaking, “known” position at time $t$ (or, better, to a
$\mathcal{F}_{t}$-measurable r.v.) is exactly the opposite of such position.
$\mathcal{F}_{t}$-1-cash-additivity and $\mathcal{F}_{t}$-cash-additivity have
a similar interpretation to cash-additivity for dynamic risk measures; we
stress that for $\mathcal{F}_{t}$-1-cash-additivity the translation has an
impact only on the first variable, i.e., on the sub-portfolio. Full allocation
states that, at any time $t$, the capital requirement $\Lambda_{t}(Y;Y)$ is
fully divided into the different sub-portfolios. As emphasized by Kalkbrener
[27] in the static case, however, full allocation and no-undercut together
with $\Lambda(X;X)=\rho(X)$ are incompatible for convex risk measures that are
not coherent since these axioms together imply subadditivity. Nevertheless, as
underlined by Brunnermeier and Cheridito [8], full allocation can be dropped
when, e.g., the CAR is considered only for monitoring purposes (see also [10,
11] and the references therein for a deeper discussion). For the reason above,
sub-allocation and weak convexity can be defined and investigated as
alternatives to full allocation. In particular, sub-allocation implies that
the excess
$\Lambda_{t}(\sum_{i=1}^{n}Y_{i};Y)-\sum_{i=1}^{n}\Lambda_{t}(Y_{i};Y)\geq 0$
can be seen as an undivided deposit/cost that represents an extra-security
margin and can be motivated because of some costs shared by all the sub-
portfolios. Weak convexity, instead, represents a sort of convexity in the
first variable.

While the previous axioms are simply a reformulation in a dynamic setting of
those required in the static case, the following are specific for dynamic CARs
and, up to our knowledge, have not been introduced yet in the literature of
CARs. In particular, time-consistency is specific for the dynamic setting and
represents a recursivity requirement, similarly to the one imposed to dynamic
risk measures.

\- time-consistency of type 1:
$\Lambda_{s}(-\Lambda_{t}(X;Y);Y)=\Lambda_{s}(X;Y)$ for any $X,Y\in
L^{\infty}(\mathcal{F}_{T})$ and $0\leq s\leq t\leq T$.

\- time-consistency of type 2:
$\Lambda_{s}(-\Lambda_{t}(X;Y);-\rho_{t}(Y))=\Lambda_{s}(X;Y)$ for any $X,Y\in
L^{\infty}(\mathcal{F}_{T})$ and $0\leq s\leq t\leq T$.

More specifically, time-consistency of type 2 means that the capital to be
allocated at time $s$ for $X$ as a sub-portfolio of $Y$ should be the same as
the capital to be allocated at time $s$ for the sub-position
$-\Lambda_{t}(X;Y)$ of the whole position $-\rho_{t}(Y)$, arising from a
intermediate step from $T$ to $t$; that is, by considering two consecutive
steps from $T$ to $t$ and then from $t$ to $s$, where in the last step we
don’t consider the final position $Y$ but its margin at time $t$. Time-
consistency of type 1, instead, fixes the whole portfolio whatever is the
evaluation time and represents a recursivity property guaranteeing that the
capital to be allocated at $s$ to $X$ as a sub-portfolio of $Y$ is the same as
the capital to be allocated when we proceed backward in time in two steps and
when the whole position $Y$ is fixed.

We now investigate the relation between dynamic CARs and dynamic convex risk
measures and, in particular, the existence of dynamic CARs satisfying some
suitable axioms among those listed.

In the following, let $(\rho_{t})_{t\in[0,T]}$ be a dynamic convex risk
measure satisfying time-consistency and assume that, for any $t\in[0,T]$,

###### Assumption 3

$\rho_{t}(X)=\operatorname*{ess\,max}_{Q\in\mathcal{Q}_{t}}\\{E_{Q}[-X|\mathcal{F}_{t}]-c_{t}(Q)\\}\quad\mbox{for
any }X\in L^{\infty}(\mathcal{F}_{T}),$ (5)

for the minimal penalty term $c_{t}$ defined in (2) and $\mathcal{Q}_{t}$
defined accordingly.

We recall that Assumption 3 is fulfilled, for instance, when $\rho_{t}$ is
continuous from below (see Bion-Nadal [4]).

We recall from the static case that the gradient and the subdifferential CARs
are related to the optimal scenarios in the dual representation of
$\rho=\rho_{0}$ (see [15, 11, 27]). Similarly to the static case (see [11]), a
dynamic counterpart of the subdifferential CAR can be defined as follows:

$\Lambda^{sub}_{t}(X;Y)\triangleq
E_{Q_{t}^{Y}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{Y}),$ (6)

where

$Q_{t}^{Y}\in\arg\operatorname*{ess\,max}_{Q\in\mathcal{Q}_{t}}\\{E_{Q}[-Y|\mathcal{F}_{t}]-c_{t}(Q)\\}.$
(7)

Note that, similarly to the static case, the CAR above is not uniquely
assigned but represents a family of CARs depending on the choice of
$Q_{t}^{Y}$ with respect to it is defined. A discussion on this point will
follow later.

Similarly as in [15], where the static version is considered, we define the
subdifferential of $\rho_{t}$ at $X\in L^{\infty}(\mathcal{F}_{T})$ as

$\displaystyle\partial\rho_{t}(X)\triangleq\left\\{Q\in\mathcal{Q}_{t}:\ \
\rho_{t}(Y)\geq\rho_{t}(X)+E_{Q}[-(Y-X)|\mathcal{F}_{t}]\ \text{ for all }Y\in
L^{\infty}(\mathcal{F}_{T})\right\\}.$

It is said (see, e.g., [15, 44]) that $\rho_{t}$ is subdifferentiable at $X\in
L^{\infty}(\mathcal{F}_{T})$ if $\partial\rho_{t}(X)$ is non-empty.

The following lemma extends to the dynamic case the relationship between
subdifferential and optimal scenarios of risk measures, well known in the
static case (see, among others, Delbaen [15], Ruszczyński and Shapiro [40],
and Zălinescu [44]).

###### Lemma 4

For a dynamic convex risk measure $\rho_{t}$ satisfying Assumption 3,
$\partial\rho_{t}(X)$ coincides with the set formed by all the optimal
scenarios in (5) (hence $\partial\rho_{t}(X)\neq\emptyset$ for any $X\in
L^{\infty}(\mathcal{F}_{T})$). Moreover, the optimal scenario $Q_{t}^{X}$ is
uniquely determined whenever $\rho_{t}(X)$ is Gateaux differentiable at $X$.

Proof. Let us observe that (5) guarantees that there exists at least one
element in $\partial\rho_{t}(X)$. In fact, in this case, for every
$Q_{t}^{X}\in{\rm
arg\,}\operatorname*{ess\,max}_{Q\in\mathcal{Q}_{t}}\left\\{E_{{Q}_{t}}[-X|\mathcal{F}_{t}]-c_{t}(Q)\right\\}$

$\displaystyle\rho_{t}(Y)-\rho_{t}(X)$
$\displaystyle=\operatorname*{ess\,max}_{Q\in\mathcal{Q}_{t}}\left\\{E_{{Q}_{t}}[-Y|\mathcal{F}_{t}]-c_{t}(Q)\right\\}-\left[E_{Q_{t}^{X}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{X})\right]$
$\displaystyle\geq E_{Q_{t}^{X}}[-(Y-X)|\mathcal{F}_{t}]$

holds for any $Y\in L^{\infty}(\mathcal{F}_{T})$. Thus $Q_{t}^{X}$ belongs to
$\partial\rho_{t}(X)$.

Vice versa, $Q_{t}^{X}\in\partial\rho_{t}(X)$ implies that it is an optimal
scenario in the dual representation of $\rho_{t}(X)$. Indeed,
$Q_{t}^{X}\in\partial\rho_{t}(X)$ implies that
$E_{Q_{t}^{X}}[-Y|\mathcal{F}_{t}]-\rho_{t}(Y)\leq
E_{Q_{t}^{X}}[-X|\mathcal{F}_{t}]-\rho_{t}(X)$ for any $Y\in
L^{\infty}(\mathcal{F}_{T})$, hence

$c_{t}(Q_{t}^{X})={\operatorname*{ess\,max}}_{Y\in
L^{\infty}(\mathcal{F}_{T})}\\{E_{Q_{t}^{X}}[-Y|\mathcal{F}_{t}]-\rho_{t}(Y)\\}=E_{Q_{t}^{X}}[-X|\mathcal{F}_{t}]-\rho_{t}(X).$

It then follows that

$\rho_{t}(X)=E_{Q_{t}^{X}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{X}),$

therefore $Q_{t}^{X}\in\partial\rho_{t}(X)$ is an optimal scenario in the dual
representation of $\rho_{t}(X)$.

As in the static case, we now show that $\partial\rho_{t}(X)$ is a singleton
when $\rho_{t}$ is Gateaux differentiable. To this aim, we say (see, e.g.,
[40, 44] for the static case) that $\rho_{t}$ is Gateaux differentiable at $X$
if $\rho_{t}$ is directionally differentiable at $X$ in the direction $Y$,
i.e., if there exists the limit in

$D\rho_{t}(X;Y)\triangleq\lim_{h\to 0}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h},$
(8)

and, for any $X\in{\rm dom}(\rho_{t})$, the directional derivative.

$D\rho_{t}(X;Y)=\lim_{h\searrow
0^{+}}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h}\geq\lim_{h\searrow
0^{+}}\frac{E_{Q_{t}^{X}}[-hY|\mathcal{F}_{t}]}{h}=E_{Q_{t}^{X}}[-Y|\mathcal{F}_{t}],$

while

$D\rho_{t}(X;Y)=\lim_{h\nearrow
0^{-}}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h}\leq\lim_{h\nearrow
0^{-}}\frac{E_{Q_{t}^{X}}[-hY|\mathcal{F}_{t}]}{h}=E_{Q_{t}^{X}}[-Y|\mathcal{F}_{t}].$

From the uniqueness of the limit in (8) we can conclude that

$D\rho_{t}(X;Y)=E_{Q_{t}^{X}}[-Y|\mathcal{F}_{t}].$

Consequently, for any pair $Q^{1}_{X},Q^{2}_{X}\in{\rm
arg\,}\operatorname*{ess\,max}_{\mathcal{Q}_{t}}\left\\{E_{Q}[-X|\mathcal{F}_{t}]-c_{t}(Q)\right\\}$
the identity

$E_{Q^{1}_{X}}[-Y|\mathcal{F}_{t}]=E_{Q_{X}^{2}}[-Y|\mathcal{F}_{t}]$

holds for every $Y\in L^{\infty}(\mathcal{F}_{T})$. Thus $Q^{1}_{X}$ and
$Q^{2}_{X}$ coincide on $\mathcal{F}_{T}$ and $D\rho_{t}(X;Y)$ can be
represented by (one of) them. Hence, $\partial\rho_{t}(X)$ is a singleton if
$\rho_{t}$ is Gateaux differentiable at $X$.

Note that, from the arguments above, the dynamic subdifferential CAR
$\Lambda^{sub}_{t}$ can be also written as

$\Lambda^{sub}_{t}(X;Y)=\rho_{t}(Y)-E_{Q_{t}^{Y}}[-(Y-X)|\mathcal{F}_{t}].$
(9)

The previous lemma implies that, for a dynamic convex risk measure satisfying
Assumption 3, the subdifferential CAR represents a family of CARs since, in
general, there may be several optimal scenario $Q_{t}^{X}$. The optimal
scenario $Q_{t}^{X}$ is instead uniquely determined whenever $\rho_{t}(X)$ is
Gateaux differentiable at $X$.

The non-differentiability of a risk measure (leading to a
$\partial\rho_{t}(X)$ that is not a singleton in general, hence to a family of
-static or dynamic- CARs) could seem to be problematic since different CARs
may rank differently sub-portfolios. However, we wish to emphasize that this
may happen in general since to any (static or dynamic) risk measure one can
associate different capital allocation rules according to different axioms
(e.g., gradient or subdifferential, marginal, quantile-based, Aumann-Shapley,
…). See [21, 41, 43]. In order to define uniquely a CAR (e.g., out of a family
of CARs), one possibility could be to choose an optimal capital allocation
rule where optimality is taken with respect to a suitable deviation measure
(as discussed by Dhaene et al. [21]). Another possibility could be to fix the
family of CARs according to some desirable axioms and, in case of non-
differentiability of the risk measure and of subdifferential CARs, to choose
the optimal scenario on which the CAR is built according to some criteria,
e.g. of minimal martingale or entropy (see Föllmer and Schweizer [24],
Frittelli [25], Centrone and Rosazza Gianin [11] for a discussion on related
CARs).

The following result establishes a one-to-one correspondence between
$(\Lambda_{t})_{t\in[0,T]}$ and $(\rho_{t})_{t\in[0,T]}$ and the relationship
between the properties of $(\Lambda_{t})_{t\in[0,T]}$ and of
$(\rho_{t})_{t\in[0,T]}$, in line with Theorem 4.3 of Kalkbrener [27] and of
Proposition 4 of Centrone and Rosazza Gianin [11].

###### Proposition 5

a) If $\Lambda_{t}:L^{\infty}(\mathcal{F}_{T})\times
L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$, for any
$t\in[0,T]$, is a monotone and weakly convex functional satisfying
$\Lambda_{t}(X;Y)\leq\Lambda_{t}(X;X)$ for any $X,Y\in
L^{\infty}(\mathcal{F}_{T})$ (no-undercut) and time-consistency of type 1
(resp. of type 2), then the associated dynamic risk measure $\rho_{t}$,
defined as $\rho_{t}(X)\triangleq\Lambda_{t}(X;X)$ for $X\in
L^{\infty}(\mathcal{F}_{T})$, is a monotone, convex and weak time-consistent
(resp. time-consistent) risk measure. Moreover, if $\Lambda_{t}$ satisfies
also $\mathcal{F}_{t}$-cash-additivity, then the associated risk measure
$\rho_{t}$ is also cash-additive (hence, a dynamic convex and time-consistent
risk measure).

b) If $\rho_{t}$ is a dynamic convex risk measure satisfying continuity from
below, then there exists at least a dynamic CAR $\Lambda_{t}$ (e.g.,
$\Lambda^{sub}_{t}$) satisfying monotonicity, weakly convexity,
$\mathcal{F}_{t}$-1-cash-additivity and no-undercut.

Proof. a) Monotonicity of $\rho_{t}$ can be checked easily due to the
corresponding properties of $\Lambda_{t}$.

If $\Lambda_{t}$ satisfies time-consistency of type 1, then weak time-
consistency of $\rho_{t}$ follows. Indeed, it holds that for any $0\leq s\leq
t\leq T$ and any $X\in L^{\infty}(\mathcal{F}_{T})$

$\rho_{s}(X)=\Lambda_{s}(X;X)=\Lambda_{s}(-\Lambda_{t}(X;X);X)\leq\rho_{s}(-\rho_{t}(X)),$

where the second equality comes from time-consistency of type 1, while the
inequality follows from the no-undercut property.

Time-consistency of type 2 of $\Lambda_{t}$, instead, implies time-consistency
of $\rho_{t}$. Indeed, for any $0\leq s\leq t\leq T$ and any $X,Y\in
L^{\infty}(\mathcal{F}_{T})$ we have

$\rho_{s}(X)=\Lambda_{s}(X;X)=\Lambda_{s}(-\Lambda_{t}(X;X);-\rho_{t}(X))=\rho_{s}(-\rho_{t}(X)),$

where the second equality comes from time-consistency of type 2.

Convexity: from $\rho_{t}(X)\triangleq\Lambda_{t}(X;X)$ it follows that for
any $t\in[0,T]$, $\alpha\in[0,1]$ and $X,Y\in L^{\infty}(\mathcal{F}_{T})$

$\displaystyle\rho_{t}(\alpha X+(1-\alpha)Y)$ $\displaystyle=$
$\displaystyle\Lambda_{t}(\alpha X+(1-\alpha)Y;\alpha X+(1-\alpha)Y)$
$\displaystyle\leq$ $\displaystyle\alpha\Lambda_{t}(X;\alpha
X+(1-\alpha)Y)+(1-\alpha)\Lambda_{t}(Y;\alpha X+(1-\alpha)Y)$
$\displaystyle\leq$ $\displaystyle\alpha\rho_{t}(X)+(1-\alpha)\rho_{t}(Y),$

where the former inequality is due to weakly convexity of $\Lambda_{t}$, while
the latter from no-undercut.

Cash-additivity of $\rho_{t}$ is straightforward under $\mathcal{F}_{t}$-cash-
additivity of $\Lambda_{t}$.

b) By the hypothesis on $\rho_{t}$, the dual representation (5) holds true
(see Bion-Nadal [4] and the references therein). Consider now

$\Lambda_{t}(X;Y)=\Lambda^{sub}_{t}(X;Y)=E_{Q_{t}^{Y}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{Y}),$

where $Q_{t}^{Y}$ denotes an optimal scenario in the dual representation of
$\rho_{t}(Y)$.

By definition of $Q_{t}^{Y}$, $\Lambda_{t}^{sub}(Y;Y)=\rho_{t}(Y)$ and
$\Lambda^{sub}_{t}$ is a dynamic CAR. Monotonicity, weakly convexity and
$\mathcal{F}_{t}$-1-cash-additivity can be easily checked. No-undercut,
instead, follows by

$\Lambda_{t}(X;Y)=E_{Q_{t}^{Y}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{Y})\leq\operatorname*{ess\,max}_{Q\in\mathcal{Q}_{t}}\\{E_{Q}[-X|\mathcal{F}_{t}]-c_{t}(Q)\\}=\rho_{t}(X)$

for any $t\in[0,T]$ and $X,Y\in L^{\infty}(\mathcal{F}_{T})$. Furthermore,
sub-allocation is also fulfilled. Indeed, for any $Y_{1},...,Y_{n},Y\in
L^{\infty}(\mathcal{F}_{T})$ such that $Y=\sum_{i=1}^{n}Y_{i}$ and $t\in[0,T]$
it holds that

$\displaystyle\Lambda^{sub}_{t}\left(\sum_{i=1}^{n}Y_{i};Y\right)$
$\displaystyle=$ $\displaystyle
E_{Q_{t}^{Y}}\left[\left.-\sum_{i=1}^{n}Y_{i}\right|\mathcal{F}_{t}\right]-c_{t}(Q_{t}^{Y})$
$\displaystyle\geq$
$\displaystyle\sum_{i=1}^{n}E_{Q_{t}^{Y}}[-Y_{i}|\mathcal{F}_{t}]-nc_{t}(Q_{t}^{Y})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\\{E_{Q_{t}^{Y}}[-Y_{i}|\mathcal{F}_{t}]-c_{t}(Q_{t}^{Y})\\}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\Lambda^{sub}_{t}(Y_{i};Y).$

Note that, starting from a dynamic and time-consistent CAR, we can obtain a
time-consistent dynamic risk measure. The converse implication, instead, needs
a further investigation. A further step will be then to investigate the
existence of a time-consistent CAR induced by a time-consistent dynamic risk
measure. For a $\rho_{t}$ coming from a $g$-expectation we will show in the
next section that the existence is guaranteed.

##  4 CARs associated to a $\rho_{t}$ induced by a $g$-expectation

In the following, we will restrict our attention to dynamic risk measures that
are induced by $g$-expectations. The main motivations for this choice can be
summarized as follows. First, as shown in [14, 35, 39], a wide family of
dynamic time-consistent convex risk measures satisfying some further
assumptions come from a $g$-expectation. Second and somehow related to the
previous point, there is a huge literature on risk measures induced by BSDEs
in a Brownian setting or in a setting with jumps. See, among others, Barrieu
and El Karoui [3], Delbaen et al. [18], Rosazza Gianin [39], Laeven and Stadje
[31], Quenez and Sulem [36], Calvia and Rosazza Gianin [9]. Finally, some
recent works (see Kromer and Overbeck [29, 30] and Mabitsela et al. [32])
already focus on dynamic risk measures induced by BSDEs and Volterra equations
to investigate the gradient allocation.

Let $(\rho_{t})_{t\in[0,T]}$ be a dynamic convex and time-consistent risk
measure that is induced by a $g$-expectation in a Brownian setting, i.e.,
$(\rho_{t}(X),Z_{t}^{X})$ solves the following BSDE

$\rho_{t}(X)=-X+\int_{t}^{T}g_{\rho}(s,Z_{s}^{X})\,ds-\int_{t}^{T}Z_{s}^{X}\,dB_{s},\quad
X\in L^{\infty}(\mathcal{F}_{T}),$ (10)

for a suitable driver $g_{\rho}$ satisfying the usual assumptions and
convexity in $z$. In particular, $g_{\rho}$ is uniformly Lipschitz in $z$ and
$g_{\rho}(s,0)=0$ for any $s\in[0,T]$.

Inspired by Kromer and Overbeck [29] where the authors proved that, under
Gateaux differentiability of $\rho_{t}$, the gradient CAR for a $\rho_{t}$ as
in (10) satisfies a BSDE with a driver depending on the gradient of
$g_{\rho}$, we now introduce a general formulation of dynamic CARs also
induced by a $g$-expectation but going beyond the gradient approach and under
weaker assumptions than differentiability. Assume now that also the dynamic
CAR $\Lambda_{t}$ we are looking for is induced by a $g$-expectation with a
different driver $g_{\Lambda}$. More precisely, assume that

$\Lambda_{t}(X;Y)=-X+\int_{t}^{T}g_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})\,ds-\int_{t}^{T}Z_{s}^{X,Y}\,dB_{s},$
(11)

where $Z_{s}^{X,Y}$ is part of the solution while $Z_{s}^{Y}$ comes from

$\rho_{t}(Y)=-Y+\int_{t}^{T}g_{\rho}(s,Z_{s}^{Y})\,ds-\int_{t}^{T}Z_{s}^{Y}\,dB_{s},$

and with $g_{\Lambda}(s,z,z^{y})$ satisfying (for any
$z^{y}\in\mathbb{R}^{d}$) the usual non-normalized assumptions on $z$ and the
condition

$g_{\Lambda}(s,z,z)=g_{\rho}(s,z)\quad\mbox{for any
}s\in[0,T],z\in\mathbb{R}^{d}.$ (12)

In this case, the driver $g_{\Lambda}(s,z,z^{y})$ depends then on an
additional parameter $z^{y}$. Note that the assumptions on $g_{\Lambda}$ imply
the existence and uniqueness of the solution
$(\Lambda_{t}(X;Y);Z_{t}^{X,Y})_{t\in[0,T]}$ of (11), while condition (12)
guarantees that $\Lambda_{t}$ is a dynamic CAR, i.e.,
$\Lambda_{t}(Y;Y)=\rho_{t}(Y)$ for any $t\in[0,T]$ and $Y\in
L^{\infty}(\mathcal{F}_{T})$.

Given a dynamic risk measure $(\rho_{t})_{t\in[0,T]}$ induced by a
$g_{\rho}$-expectation, it is then possible to define several dynamic capital
allocations induced by a $g_{\Lambda}$-expectation with $g_{\Lambda}$
fulfilling condition (12). Viceversa, given a dynamic family
$(\Lambda_{t})_{t\in[0,T]}$ induced by a $g_{\Lambda}$-expectation, the
associated dynamic risk measure is uniquely determined via (12).

As already discussed above, the assumption of a dynamic CAR induced by a
$g$-expectation generalizes the gradient case and, in view of a result of
Coquet et al. [14] (see also Remark 6 below), seems to be rather reasonable
for risk measures coming from $g$-expectations.

###### Remark 6

a) If $(\rho_{t})_{t\in[0,T]}$ is Gateaux differentiable at any time
$t\in[0,T]$, then the gradient allocation $(\Lambda_{t}^{grad})_{t\in[0,T]}$
is of the form (11) with

$g_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})=\nabla g_{\rho}(s,Z_{s}^{Y})\cdot
Z_{s}^{X,Y},$ (13)

where, in the $d$-dimensional case, $\nabla g_{\rho}$ stands for the gradient
of $g_{\rho}$ with respect to $z$. We observe that this result has been
already proved in [29, Theorem 3.1] for BSDEs where $\nabla
g_{\rho}(s,Z_{s}^{Y})$ is from a BMO444Note that in [29] $g_{\rho}$ may have
quadratic growth in $z$, hence $\nabla g_{\rho}$ is not bounded in general.
$\nabla g_{\rho}(s,Z_{s}^{Y})$ is then assumed to be from a BMO in order to be
able to apply Girsanov Theorem and to define an equivalent probability measure
in terms of $\nabla g_{\rho}$. For a precise definition of BMO we refer to
Kazamaki [28]..

b) It seems quite reasonable to assume that a dynamic CAR associated to a
dynamic risk measure as in (10) is induced by a $g_{\Lambda}$-expectation.
Indeed, let $(\rho_{t})_{t\in[0,T]}$ be as in (10) and let
$(\Lambda_{t})_{t\in[0,T]}$ be convex, monotone and cash-additive in $X$, and
satisfying time-consistency. Assume, in addition, that $\Lambda_{t}$ satisfies
$\mathcal{E}_{\bar{\mu}}$-dominance for some $\bar{\mu}>0$, that is,

$\Lambda_{t}(X;Y)-\Lambda_{t}(Z;Y)\leq\mathcal{E}_{\bar{\mu}}(-(X-Z)|\mathcal{F}_{t}),\quad\mbox{
for any }X,Y,Z\in L^{\infty}(\mathcal{F}_{T}),t\in[0,T].$

It then follows by Coquet et al. [14] that $(\Lambda_{t})_{t\in[0,T]}$ is also
induced by a $\bar{g}$-expectation for some $\bar{\mu}$-Lipschitz driver
$\bar{g}=\bar{g}_{Y}$, i.e., there exists some
$\bar{g}_{Y}(s,z)\leq\bar{\mu}|z|$ such that
$\Lambda_{t}(X;Y)=\mathcal{E}_{\bar{g}_{Y}}(-X|\mathcal{F}_{t})$ for any
$X,Y\in L^{\infty}(\mathcal{F}_{T})$.

Note that $\mathcal{E}_{\mu}$-dominance of $(\Lambda_{t})_{t\in[0,T]}$ is
guaranteed, for instance, when

$\Lambda_{t}(X;Y)-\Lambda_{t}(Z;Y)\leq\rho_{t}(X-Z),\quad\mbox{ for any
}X,Y,Z\in L^{\infty}(\mathcal{F}_{T}),t\in[0,T].$

The condition above seems to be a reasonable generalization of

$\Lambda_{t}(X;Y)\leq\rho_{t}(X)=\mathcal{E}_{g_{\rho}}(-X|\mathcal{F}_{t})\leq\mathcal{E}_{\mu}(-X|\mathcal{F}_{t}),\quad\text{for
any }X,Y\in L^{\infty}(\mathcal{F}_{T}),t\in[0,T],$

that is automatically fulfilled under no-undercut.

Assume now that $\rho_{t}$ is convex but non necessarily Gateaux
differentiable and assume that $\partial\rho_{t}(X)\neq\emptyset$, that is,
$\rho_{t}$ is subdifferentiable at $X$ (see Lemma 4 and [40] for a detailed
discussion). As from Lemma 4, we can identify an element of
$\partial\rho_{t}(X)$ with the corresponding probability measure $Q$.

We are now ready to prove the existence of a dynamic CAR that is time-
consistent for a dynamic time-consistent risk measure induced by a
$g$-expectation, as a follow-up of Proposition 5-b).

With an abuse of notation, formulations containing the term $\partial
g_{\rho}$ should be read as follows: to any element of the set $\partial
g_{\rho}$ we associate an element (e.g., of $\partial\rho_{t}(X)$,
$\Lambda_{t}^{sub}$ or $g_{\Lambda}$). To be more precise, in (14) to any
element of the family $(\partial g_{\rho}(u,Z_{u}^{X}))_{u\in[0,t]}$ of the
subdifferential of $g_{\rho}$ it corresponds an element $Q_{t}^{X}$ of the
subdifferential $\partial\rho_{t}(X)$ and, consequently, a different element
of the family $\Lambda^{sub}$. Or, better, similarly to the general case,
$\Lambda^{sub}$ is not uniquely defined but corresponds to a family of CARs
depending on the choice of the optimal scenario to which is associated. A
discussion on the subdifferentiability of $g_{\rho}$ is postponed to Remark
13.

######  Proposition 7 (Existence of a CAR satisfying time-consistency)

Let $(\rho_{t})_{t\in[0,T]}$ be a dynamic convex risk measure that is induced
by a $g_{\rho}$-expectation as in (10).

If $g_{\rho}$ is subdifferentiable, then $\rho_{t}$ is also subdifferentiable
and elements $Q_{t}^{X}$ defined by

$E\left[\left.\frac{dQ_{t}^{X}}{dP}\right|\mathcal{F}_{t}\right]\triangleq\mathcal{E}(\partial
g_{\rho}(t,Z_{t}^{X})\cdot B)=\exp\left\\{-\frac{1}{2}\int_{0}^{t}\|\partial
g_{\rho}(u,Z_{u}^{X})\|^{2}\,du-\int_{0}^{t}\partial
g_{\rho}(u,Z_{u}^{X})\,dB_{u}\right\\}$ (14)

belong to $\partial\rho_{t}(X)$.

Furthermore, $\Lambda^{sub}_{t}(X;Y)$ related to $Q_{t}^{X}$ as above is a
(family of) time-consistent dynamic CAR(s) satisfying monotonicity, no-
undercut and sub-allocation. Existence of a CAR satisfying the axioms above is
therefore guaranteed.

Moreover, $\Lambda^{sub}_{t}(X;Y)$ satisfies the following BSDE

$\Lambda^{sub}_{t}(X;Y)=-X+\int_{t}^{T}\left[g_{\rho}(s,Z_{s}^{Y})+\partial
g_{\rho}(s,Z_{s}^{Y})\cdot(Z_{s}^{X,Y}-Z_{s}^{Y})\right]\,ds-\int_{t}^{T}Z_{s}^{X,Y}\,dB_{s}.$
(15)

Proof. Let $g_{\rho}$ be subdifferentiable and convex and let
$\bar{q}\in\partial g_{\rho}(s,z)$, where $\partial
g_{\rho}(s,z)\triangleq\\{q\in\mathbb{R}^{d}:g_{\rho}(s,u)\geq
g_{\rho}(s,z)+q\cdot(u-z)\mbox{ for any }u\in\mathbb{R}^{d}\\}$. It then holds
that $g_{\rho}$ is continuous in the second variable and
$g_{\rho}(s,z)=\bar{q}\cdot z-g^{*}_{\rho}(s,\bar{q})$ where $g^{*}_{\rho}$ is
the convex conjugate of $g_{\rho}$. Note that any element $\bar{q}\in\partial
g_{\rho}(s,z)$ is such that $\|\bar{q}\|\leq\mu$ where $\mu>0$ is the
Lipschitz constant of $g_{\rho}$. This follows because
$g^{*}_{\rho}(s,\bar{q})=+\infty$ for $\|\bar{q}\|>\mu$ (see, e.g., [18],
Prop. 3.6) and $g_{\rho}(s,z)=\bar{q}\cdot z-g^{*}_{\rho}(s,\bar{q})$ is
finite. In the following, we sometimes denote $g=g_{\rho}$ when there is no
possible misunderstanding. With a slight abuse of notation, we also indicate
an element in the subdifferential of $g(s,z)$ by $\partial g(s,z)$.

We start proving that $\rho_{t}$ is also subdifferentiable and that
$Q_{t}^{X}$ in (14) belongs to $\partial\rho_{t}(X)$. Fix now $X\in
L^{\infty}(\mathcal{F}_{T})$ and consider any $Y\in
L^{\infty}(\mathcal{F}_{T})$. By (10),

$\displaystyle\rho_{t}(Y)-\rho_{t}(X)$ $\displaystyle=$
$\displaystyle-(Y-X)+\int_{t}^{T}\left[g_{\rho}(s,Z_{s}^{Y})-g_{\rho}(s,Z_{s}^{X})\right]\,ds-\int_{t}^{T}\left(Z_{s}^{Y}-Z_{s}^{X}\right)\,dB_{s}$
$\displaystyle\geq$ $\displaystyle-(Y-X)+\int_{t}^{T}\partial
g(s,Z_{s}^{X})\cdot\left(Z_{s}^{Y}-Z_{s}^{X}\right)\,ds-\int_{t}^{T}\left(Z_{s}^{Y}-Z_{s}^{X}\right)\,dB_{s}$
$\displaystyle\geq$
$\displaystyle-(Y-X)-\int_{t}^{T}\left(Z_{s}^{Y}-Z_{s}^{X}\right)\,d\bar{B}^{Q^{X}}_{s}$

where $d\bar{B}^{Q^{X}}_{s}=dB_{s}-\partial g(s,Z_{s}^{X})\,ds$ and, by
Girsanov Theorem, $(\bar{B}^{Q^{X}}_{t})_{t\in[0,T]}$ is a $Q^{X}$-Brownian
motion. By taking the conditional expectation with respect to $Q_{t}^{X}$ of
the first and last term in the chain of inequalities above, it follows that

$\rho_{t}(Y)-\rho_{t}(X)\geq
E_{Q_{t}^{X}}[-(Y-X)|\mathcal{F}_{t}]\quad\mbox{for any }Y\in
L^{\infty}(\mathcal{F}_{T}),$

hence $\rho_{t}$ is subdifferentiable at $X$ and
$Q_{t}^{X}\in\partial\rho_{t}(X)$ is an optimal scenario.

It is easy to check that if $\rho_{t}$ is subdifferentiable at $X\in
L^{\infty}(\mathcal{F}_{T})$, then $\rho_{t}$ is continuous from above at $X$.
By the arguments above and by the characterization of the penalty term in
Delbaen et al. [18], Theorem 3.2, and Barrieu and El Karoui [3], Theorem 7.4,
it then follows that

$\displaystyle\rho_{t}(X)$ $\displaystyle=$ $\displaystyle
E_{Q_{t}^{X}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{X})$ $\displaystyle=$
$\displaystyle
E_{Q_{t}^{X}}[-X|\mathcal{F}_{t}]-E_{Q_{t}^{X}}\left[\left.\int_{t}^{T}g^{*}_{\rho}(s,\partial
g(s,Z_{s}^{X}))\,ds\right|\mathcal{F}_{t}\right]$ $\displaystyle=$
$\displaystyle E_{Q_{t}^{X}}\left[\left.-X-\int_{t}^{T}\left[\partial
g(s,Z_{s}^{X})\cdot
Z_{s}^{X}-g(s,Z_{s}^{X})\right]\,ds\right|\mathcal{F}_{t}\right].$

By similar arguments and by the martingale representation theorem, there
exists a unique stochastic process $(Z_{s}^{X,Y})_{s\in[0,T]}$ such that

$\displaystyle\Lambda^{sub}_{t}(X;Y)$ $\displaystyle=$ $\displaystyle
E_{Q_{t}^{Y}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{Y})$ $\displaystyle=$
$\displaystyle E_{Q_{t}^{Y}}\left[-X-\int_{t}^{T}\left[\partial
g(s,Z_{s}^{Y})\cdot
Z_{s}^{Y}-g(s,Z_{s}^{Y})\right]\,ds|\mathcal{F}_{t}\right]$ $\displaystyle=$
$\displaystyle-X-\int_{t}^{T}\left[\partial g(s,Z_{s}^{Y})\cdot
Z_{s}^{Y}-g(s,Z_{s}^{Y})\right]\,ds-\int_{t}^{T}Z_{s}^{X,Y}\,d\bar{B}^{Q^{Y}}_{s}$
$\displaystyle=$ $\displaystyle-X+\int_{t}^{T}\left[\partial
g(s,Z_{s}^{Y})\cdot(Z_{s}^{X,Y}-Z_{s}^{Y})+g(s,Z_{s}^{Y})\right]\,ds-\int_{t}^{T}Z_{s}^{X,Y}\,dB_{s},$

hence $(\Lambda^{sub}_{t}(X;Y),Z_{t}^{X,Y})$ solves a
$g_{\Lambda}$-expectation with final condition $-X$ and driver

$g_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})=\partial
g(s,Z_{s}^{Y})\cdot(Z_{s}^{X,Y}-Z_{s}^{Y})+g(s,Z_{s}^{Y})$

depending on $Z_{s}^{Y}$. It is straightforward to check that
$g_{\Lambda}(s,0,Z_{s}^{Y})=g(s,Z_{s}^{Y})-\partial g(s,Z_{s}^{Y})\cdot
Z_{s}^{Y}$ satisfies $(g_{2})$ since $g$ satisfies condition (4) and, for any
$s\in[0,T]$, $z\in\mathbb{R}^{d}$, $q\in\partial g(s,z)$ is such that
$\|q\|\leq\mu$ where $\mu>0$ is the Lipschitz constant of $g$. We now prove
time-consistency of type 1 and 2, i.e.

$\Lambda_{s}^{sub}(-\Lambda_{t}^{sub}(X;Y);Y)=\Lambda_{s}^{sub}(X;Y)\quad\text{and}\quad\Lambda_{s}^{sub}(-\Lambda_{t}^{sub}(X;Y);-\rho_{t}(Y))=\Lambda_{s}^{sub}(X;Y),$

for any $0\leq s\leq t\leq T$ and any $X,Y\in L^{\infty}$. We observe that

$\displaystyle\Lambda_{s}^{sub}(X;Y)$
$\displaystyle=-X+\int_{s}^{T}g_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})\,ds-\int_{s}^{T}Z^{X,Y}_{s}dB_{s}$
$\displaystyle=-X+\int_{t}^{T}g_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})\,ds-\int_{t}^{T}Z^{X,Y}_{s}dB_{s}$
$\displaystyle\ \ \ \
+\int_{s}^{t}g_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})\,ds-\int_{s}^{t}Z^{X,Y}_{s}dB_{s}$
$\displaystyle=\Lambda_{t}^{sub}(X;Y)+\int_{s}^{t}g_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})\,ds-\int_{s}^{t}Z^{X,Y}_{s}dB_{s}.$

Moreover, $\Lambda_{s}^{sub}(-\Lambda_{t}^{sub}(X;Y);-\rho_{t}(Y))$ solves the
following BSDE

$\displaystyle\Lambda_{s}^{sub}(-\Lambda_{t}^{sub}(X;Y);-\rho_{t}(Y))$
$\displaystyle=\Lambda_{t}^{sub}(X;Y)+\int_{s}^{t}g_{\Lambda}(s,\bar{Z}_{s},Z_{s}^{-\rho_{t}(Y)})\,ds-\int_{s}^{t}\bar{Z}_{s}dB_{s}$
$\displaystyle=\Lambda_{t}^{sub}(X;Y)+\int_{s}^{t}g_{\Lambda}(s,\bar{Z}_{s},Z_{s}^{Y})\,ds-\int_{s}^{t}\bar{Z}_{s}dB_{s}.$

where the last equality follows from $Z_{s}^{-\rho_{t}(Y)}=Z_{s}^{Y}$, which
is due to time-consistency of $(\rho_{t})_{t\in[0,T]}$.

We then conclude that
$\Lambda_{s}^{sub}(-\Lambda_{t}^{sub}(X;Y);-\rho_{t}(Y))=\Lambda_{s}^{sub}(-\Lambda_{t}^{sub}(X;Y);Y)=\Lambda_{s}^{sub}(X;Y)$,
i.e. $\Lambda_{s}^{sub}$ satisfies both time-consistency of type 1 and 2.
Monotonicity, no-undercut and sub-allocation have been already proved in
Proposition 5-b).

###### Remark 8

a) For dynamic coherent differentiable risk measures induced by
$g_{\rho}$-expectations with differentiable $g_{\rho}$, the result above
reduces to Theorem 3.1 of Kromer and Overbeck [29]. Indeed, if
$(\rho_{t})_{t\in[0,T]}$ is coherent, then the term corresponding to the
penalty term disappears. It is then easy to check that differentiability of
$g_{\rho}$ implies that also $\rho_{t}$ is Gateaux differentiable at any time
$t\in[0,T]$ and $\Lambda^{sub}_{t}(X;Y)=\Lambda^{grad}_{t}(X;Y)$ satisfies a
$g_{\Lambda}$-expectation with final condition $-X$ and driver

$g_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})=\partial g_{\rho}(s,Z_{s}^{Y})\cdot
Z_{s}^{X,Y}=\nabla g_{\rho}(s,Z_{s}^{Y})\cdot Z_{s}^{X,Y}.$

As recalled previously, in [29] $\nabla g_{\rho}(s,Z_{s}^{Y})$ is assumed to
be from a BMO because $g_{\rho}$ may have quadratic growth in $z$. In our
case, instead, the BMO assumption can be dropped since $g_{\rho}$ is Lipschitz
in $z$, hence $\partial g$ is bounded.

b)  Note that $Z_{s}^{X,Y}$ in (15) coincides with $Z_{s}^{X}$ when $X=Y$.
Indeed, for any $X\in L^{\infty}(\mathcal{F}_{T})$

$\displaystyle\Lambda^{sub}_{t}(X;X)$ $\displaystyle=$
$\displaystyle\rho_{t}(X)$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{T}g_{\rho}(s,Z_{s}^{X})\,ds-\int_{t}^{T}Z_{s}^{X}\,dB_{s}$
$\displaystyle=$
$\displaystyle-X+\int_{t}^{T}\left[g_{\rho}(s,Z_{s}^{X})-\partial
g_{\rho}(s,Z_{s}^{X})\cdot(Z_{s}^{X}-Z_{s}^{X})\right]\,ds-\int_{t}^{T}Z_{s}^{X}\,dB_{s}.$

By (15) and from the uniqueness of the solution of a BSDE, it then follows
that $Z_{s}^{X,X}=Z_{s}^{X}$.

c) An alternative proof of the previous result can be driven by using the
formulation

$\Lambda^{sub}_{t}(X;Y)=\rho_{t}(Y)-E_{Q_{t}^{Y}}[-(Y-X)|\mathcal{F}_{t}]$

and by applying the martingale representation theorem to both the terms
$E_{Q_{t}^{Y}}[-Y|\mathcal{F}_{t}]$ and $E_{Q_{t}^{Y}}[X|\mathcal{F}_{t}]$.

In the following, we provide some particular cases of dynamic CARs from a
$g_{\Lambda}$-expectation and of the corresponding drivers $g_{\Lambda}$.

###### Remark 9

Assume that $\rho_{t}$ is a dynamic convex risk measure that is induced by a
$g_{\rho}$-expectation. The gradient, subdifferential and marginal CAR are
compatible with the formulation of dynamic CARs by means of
$g_{\Lambda}$-expectations and, as discussed below, can be obtained by
choosing a suitable driver $g_{\Lambda}$. See [15, 19, 11, 27, 41, 43] for the
static versions.

_Gradient case:_ if $\rho_{t}$ is Gateaux differentiable, then Kromer and
Overbeck [29] showed that $\Lambda^{grad}_{t}(X;Y)$ solves a BSDE with driver

$g^{grad}_{\Lambda}(s,z,z^{y})=\nabla g_{\rho}(s,z^{y})\cdot z.$

_Subdifferential case:_ if $\rho_{t}$ is only subdifferentiable, then the
previous result shows that $\Lambda^{sub}_{t}(X;Y)$ solves a BSDE with driver

$g^{sub}_{\Lambda}(s,z,z^{y})=\partial
g_{\rho}(s,z^{y})\cdot(z-z^{y})+g(s,z^{y}),$

satisfying condition (12). As already pointed out previously, $\partial
g_{\rho}(s,z^{y})$ is not necessarily unique but it is assumed to be chosen
and fixed for any $s$ and $z^{y}$.

_Marginal case:_ the marginal dynamic
$\Lambda^{marg}_{t}(X;Y)=\rho_{t}(Y)-\rho_{t}(Y-X)$ solves

$\displaystyle\Lambda^{marg}_{t}(X;Y)$ $\displaystyle=$
$\displaystyle\rho_{t}(Y)-\rho_{t}(Y-X)$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{T}\left[g_{\rho}(s,Z_{s}^{Y})-g_{\rho}(s,Z_{s}^{Y-X})\right]\,ds-\int_{t}^{T}\left(Z_{s}^{Y}-Z_{s}^{Y-X}\right)\,dB_{s}$
$\displaystyle=$
$\displaystyle-X+\int_{t}^{T}g_{\Lambda}^{\color[rgb]{0,0,0}marg}(s,Z_{s}^{X,Y},Z_{s}^{Y})\,ds-\int_{t}^{T}Z_{s}^{X,Y}\,dB_{s}$

by denoting $Z^{X,Y}_{s}\triangleq Z_{s}^{Y}-Z_{s}^{Y-X}$ and
$g^{marg}_{\Lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})=g_{\rho}(s,Z_{s}^{Y})-g_{\rho}(s,Z_{s}^{Y}-Z^{X,Y}_{s})$.
The driver $g_{\Lambda}^{marg}$ satisfies condition (12) when
$g_{\rho}(s,0)=0$ holds for any $s\in[0,T]$.

In the following, we investigate under which conditions a dynamic CAR
$(\Lambda_{t})_{t\in[0,T]}$ that is induced by a $g_{\Lambda}$-expectation
satisfies some further axioms and vice versa.

###### Proposition 10

Let $g_{\Lambda}$ be a driver satisfying the usual non-normalized assumptions
and condition (12).

i) $\Lambda_{t}(X+c_{t};Y)=\Lambda_{t}(X;Y)-c_{t}$ for any $X,Y\in
L^{\infty}(\mathcal{F}_{T})$, $c_{t}\in L^{\infty}(\mathcal{F}_{t})$.

ii) if $g_{\Lambda}(s,0,z^{y})=0$ for any $s\in[0,T]$,
$z^{y}\in\mathbb{R}^{d}$, then $\Lambda_{t}(0;Y)=0$ for any $t\in[0,T]$ and
$Y\in L^{\infty}(\mathcal{F}_{T})$.

iii) if $g_{\Lambda}(s,z,z^{y})\leq g_{\rho}(s,z)$ for any $s\in[0,T]$ and
$z,z^{y}\in\mathbb{R}^{d}$, then $\Lambda_{t}$ satisfies no-undercut.

iv) $\Lambda_{t}$ satisfies monotonicity.

v) if $\sum_{i=1}^{n}g_{\Lambda}(s,z^{i},z^{y})\leq
g_{\Lambda}(s,\sum_{i=1}^{n}z^{i},z^{y})$ for any $s\in[0,T]$,
$z^{i},z^{y}\in\mathbb{R}^{d}$, then $\Lambda_{t}$ satisfies sub-allocation.

vi) if $g_{\Lambda}(s,z,z^{y})$ is convex in $z$ for any $s\in[0,T]$,
$z^{y}\in\mathbb{R}^{d}$, then $\Lambda_{t}$ satisfies weak convexity.

Moreover, the converse implications hold in iii)-iv) if
$g_{\Lambda}(s,z,z^{y})$ is continuous in $s$.

Proof. i) and ii) are straightforward thanks to the properties of
$g$-expectations with a driver independent on the $y$-component (see Peng [34]
and Rosazza Gianin [39]). In particular, ii) is due to the fact that
$(\Lambda_{t}(0;Y);Z_{t}^{0,Y})_{t}$ with $Z_{t}^{0,Y}=0$ solves the BSDE
providing $\Lambda_{t}(0;Y)$.

The proof of the sufficient conditions iii)-vi) follows by Comparison Theorem
of BSDEs (see Theorem 2.2 of El Karoui et al. [22]) and by the properties of
$g$-expectations with a driver independent on the $y$-component (see Peng [34]
and Rosazza Gianin [39]), while the last statement (concerning necessary
conditions) follows by Theorem 4.1 of Briand et al. [7].

Note that conditions in v) and in vi) are only sufficient but not necessary
for sub-allocation and weak convexity, respectively.

It is well known that for convex risk measures  that are not coherent the
gradient approach fails to satisfy no-undercut. At the level of a convex
$g_{\Lambda}$, we have indeed that the sufficient condition in iii) is not
satisfied in general because

$g^{grad}_{\Lambda}(s,z,z)=\nabla g_{\rho}(s,z)\cdot
z=g_{\rho}(s,z)+g^{*}_{\rho}(s,\nabla g_{\rho}(s,z))\geq g_{\rho}(s,z)$

for any $z\in\mathbb{R}^{d}$, where the equality holds iff
$g^{*}_{\rho}(s,\nabla g_{\rho}(s,z))=0$.

As discussed previously, a popular static CAR related to cooperative game
theory is the Aumann-Shapley CAR  (see Aumann and Shapley [2], Tsanakas [43],
Centrone and Rosazza Gianin [11]).  We now introduce the dynamic (generalized)
Aumann-Shapley CAR defined $\omega\times\omega$ as

$\Lambda_{t}^{AS}(X;Y)=\int_{0}^{1}E_{Q_{t}^{\gamma
Y}}[-X|\mathcal{F}_{t}]\,d\gamma$ (16)

and the dynamic penalized Aumann-Shapley CAR with penalty function $c$ as

$\Lambda_{t}^{c-AS}(X;Y)=\int_{0}^{1}\left[E_{Q_{t}^{\gamma
Y}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{\gamma
Y})\right]\,d\gamma=\int_{0}^{1}\Lambda^{sub}_{t}(X;\gamma Y)\,d\gamma,$ (17)

with $Q_{t}^{\cdot}$ defined as in (7).

The following result deals with the dynamic (penalized) Aumann-Shapley and its
time-consistency.

###### Proposition 11

Let $\rho_{t}$ be a dynamic time-consistent convex risk measure that is
subdifferentiable and such that the map
$G_{t}(\gamma)\triangleq\rho_{t}(\gamma X)$ is differentiable in
$\gamma\in[0,1]$ for any $X\in L^{\infty}(\mathcal{F}_{T})$.

a) $\Lambda_{t}^{AS}$ is a dynamic CAR. Furthermore,

$\Lambda_{t}^{AS}(X;Y)=E_{P}\left[\left.-\tilde{L}^{Y}(T;t)X\right|\mathcal{F}_{t}\right],$
(18)

where

$\tilde{L}^{Y}(T;t)\triangleq\int_{0}^{1}L^{\gamma
Y}(T;t)\,d\gamma=\int_{0}^{1}\frac{\mathcal{E}(\partial g(s,Z_{s}^{\gamma
Y})\cdot B)(T)}{\mathcal{E}(\partial g(s,Z_{s}^{\gamma Y})\cdot
B)(t)}\,d\gamma$ (19)

and

$L^{H}(T;t)\triangleq\frac{\frac{dQ_{T}^{H}}{dP}}{\frac{dQ_{t}^{H}}{dP}}=\exp\left\\{-\frac{1}{2}\int_{t}^{T}\|\partial
g(u,Z_{u}^{H})\|^{2}\,du-\int_{t}^{T}\partial
g(u,Z_{u}^{H})\,dB_{u}\right\\}.$

b) If $g_{\rho}$ is positively homogeneous (hence $\rho_{t}$ is coherent),
then $\Lambda_{t}^{AS}$ is time-consistent.

c) The penalized $\Lambda_{t}^{c-AS}$ is a dynamic audacious CAR (not a
dynamic CAR in general) satisfying no-undercut.

Proof.

a) We start proving that $\Lambda_{t}^{AS}$ is a CAR and later that (18) is
satisfied. This proof is in line but extends the one of Corollary 4.1 of
Kromer and Overbeck [29] (to non-differentiable risk measures and to the
expression of $\Lambda_{t}$) and of Proposition 9(a) of Centrone and Rosazza
Gianin [11] (to the dynamic case).

Let $Q_{t}^{\gamma X}\in\partial\rho_{t}(\gamma X)$. Then, by definition,

$\displaystyle G^{\prime}_{t,-}(\gamma)$ $\displaystyle=$
$\displaystyle\lim_{h\nearrow 0}\frac{\rho_{t}(\gamma X+hX)-\rho_{t}(\gamma
X)}{h}\leq E_{Q_{t}^{\gamma X}}[-X|\mathcal{F}_{t}]$ $\displaystyle
G^{\prime}_{t,+}(\gamma)$ $\displaystyle=$ $\displaystyle\lim_{h\searrow
0}\frac{\rho_{t}(\gamma X+hX)-\rho_{t}(\gamma X)}{h}\geq E_{Q_{t}^{\gamma
X}}[-X|\mathcal{F}_{t}].$

Under the assumption of differentiability of $G_{t}(\gamma)$ for every
$\gamma\in[0,1]$, it follows that
$G^{\prime}_{t,-}(\gamma)=G^{\prime}_{t,+}(\gamma)$ and hence the following
equality holds true

$\int_{0}^{1}G^{\prime}_{t,-}(\gamma)d\gamma=\int_{0}^{1}E_{Q_{t}^{\gamma
X}}[-X|\mathcal{F}_{t}]d\gamma=\int_{0}^{1}G^{\prime}_{t,+}(\gamma)d\gamma.$

By normalization of $\rho_{t}$, $\int_{0}^{1}E_{Q_{t}^{\gamma
X}}[-X|\mathcal{F}_{t}]d\gamma=G_{t}(1)-G_{t}(0)=\rho_{t}(X)$ and,
consequently, $\Lambda_{t}^{AS}$ is a dynamic CAR.

Furthermore, by Proposition 7 and (14), we know that
$E_{P}\left[\frac{dQ_{t}^{X}}{dP}\right]=\mathcal{E}(\partial
g(t,Z_{t}^{X})\cdot B)=$$=\exp\left\\{-\frac{1}{2}\int_{0}^{t}\|\partial
g(u,Z_{u}^{X})\|^{2}\,du-\int_{0}^{t}\partial
g(u,Z_{u}^{X})\,dB_{u}\right\\}$. By Fubini-Tonelli Theorem, this implies that

$\displaystyle\Lambda_{t}^{AS}(X;Y)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}E_{Q_{t}^{\gamma Y}}[-X|\mathcal{F}_{t}]\,d\gamma$
$\displaystyle=$ $\displaystyle\int_{0}^{1}E_{P}[\left.-XL^{\gamma
Y}(T;t)\right|\mathcal{F}_{t}]\,d\gamma$ $\displaystyle=$ $\displaystyle
E_{P}\left[\left.\int_{0}^{1}(-XL^{\gamma
Y}(T;t))\,d\gamma\right|\mathcal{F}_{t}\right]$ $\displaystyle=$
$\displaystyle E_{P}\left[\left.-X\int_{0}^{1}L^{\gamma
Y}(T;t)\,d\gamma\right|\mathcal{F}_{t}\right]$ $\displaystyle=$ $\displaystyle
E_{P}\left[\left.-X\tilde{L}^{Y}(T;t)\right|\mathcal{F}_{t}\right].$

b) If $g_{\rho}$ is positively homogeneous (hence $\rho_{t}$ is coherent),
then

$\displaystyle\tilde{L}^{Y}(T;t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{\mathcal{E}(\partial g(s,Z_{s}^{\gamma
Y})\cdot B)(T)}{\mathcal{E}(\partial g(s,Z_{s}^{\gamma Y})\cdot
B)(t)}\,d\gamma$ (20) $\displaystyle=$
$\displaystyle\int_{0}^{1}\exp\left\\{-\frac{1}{2}\int_{t}^{T}\|\partial
g(u,Z_{u}^{\gamma Y})\|^{2}\,du-\int_{t}^{T}\partial g(u,Z_{u}^{\gamma
Y})\,dB_{u}\right\\}\,d\gamma$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\exp\left\\{-\frac{1}{2}\int_{t}^{T}\|\partial
g(u,\gamma Z_{u}^{Y})\|^{2}\,du-\int_{t}^{T}\partial g(u,\gamma
Z_{u}^{Y})\,dB_{u}\right\\}\,d\gamma$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\exp\left\\{-\frac{1}{2}\int_{t}^{T}\|\partial
g(u,Z_{u}^{Y})\|^{2}\,du-\int_{t}^{T}\partial
g(u,Z_{u}^{Y})\,dB_{u}\right\\}\,d\gamma$ (21) $\displaystyle=$
$\displaystyle\exp\left\\{-\frac{1}{2}\int_{t}^{T}\|\partial
g(u,Z_{u}^{Y})\|^{2}\,du-\int_{t}^{T}\partial
g(u,Z_{u}^{Y})\,dB_{u}\right\\},$ (22)

where (20) is due to $Z_{s}^{\gamma Y}=Z^{\gamma}_{s}$ and (21) to $\partial
g(u,\gamma Z_{u}^{Y})=\partial g(u,Z_{u}^{Y})$, both satisfied for any
$\gamma\in[0,1]$ because of positive homogeneity of $g(u,z)$ in $z$. By (22),
it then follows that
$\tilde{L}^{Y}(T;s)=\tilde{L}^{Y}(T;t)\cdot\tilde{L}^{Y}(t;s)$ for any $s\leq
t\leq T$, hence time-consistency of $\Lambda_{t}^{AS}$.

c) No-undercut follows from

$\displaystyle\Lambda_{t}^{c-AS}(X;Y)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\left[E_{Q_{t}^{\gamma
Y}}[-X|\mathcal{F}_{t}]-c_{t}(Q_{t}^{\gamma Y})\right]d\gamma$
$\displaystyle\leq$ $\displaystyle\int_{0}^{1}\rho_{t}(X)d\gamma=\rho_{t}(X)$

for any $X,Y\in L^{\infty}(\mathcal{F}_{T}),t\in[0,T]$. Finally, from a) it
follows that

$\Lambda_{t}^{c-AS}(X;X)=\Lambda_{t}^{AS}(X;X)-\int_{0}^{1}c_{t}(Q_{t}^{\gamma
X})d\gamma=\rho_{t}(X)-\int_{0}^{1}c_{t}(Q_{t}^{\gamma X})d\gamma$

in general does not coincide with $\rho_{t}(X)$, implying that
$\Lambda_{t}^{c-AS}$ is not a dynamic CAR but only an audacious dynamic CAR.

We conclude this section with an example of different dynamic CARs associated
to the same risk measure. Some related remarks and discussions will follow.
It is worth emphasizing that, although the example below deals with
differentiable $g_{\Lambda}$ and $g_{\rho}$ (hence with Gateaux differentiable
risk measures), there are several examples of CARs induced by $g$-expectations
and that are only subdifferentiable, e.g., the one corresponding to
$g_{\rho}(s,z)=c|z|$ and $g_{\Lambda}(s,z,z^{y})=c(|z^{y}|+|z^{y}-z|)$.
Although the case below goes beyond the Lipschitz case, we have decided to
focus on it mainly because it corresponds to one among the few examples of
BSDEs where one has an explicit solution (see, e.g., [3]), but also because
the entropic risk measure is quite popular and widely used also due to its
relation to utility theory.

######  Example 12 (Entropic risk measures with different dynamic CARS)

Let $\rho_{t}$ be a dynamic entropic risk measure, i.e.,
$\rho_{t}(X)=\lambda\ln\left(E_{P}[\exp(-\frac{X}{\lambda})|\mathcal{F}_{t}]\right)$
where $\lambda>0$ denotes the risk aversion parameter. It is well known that
the dynamic entropic risk measure solves a BSDE with driver
${\color[rgb]{0,0,0}g_{\rho}}(s,z)=\frac{1}{2\lambda}\|z\|^{2}$ (see, e.g.,
[3]).

In the following, we consider and compare different dynamic CARs associated to
the entropic risk measure above by considering the approach presented before
even if $g_{\color[rgb]{0,0,0}\rho}$ is not Lipschitz in $z$.

_Dynamic gradient CAR._ Since $g_{\color[rgb]{0,0,0}\rho}$ is differentiable
in $z$, the dynamic gradient CAR is well-defined and corresponds to

$\Lambda_{t}^{grad}(X;Y)=E_{P}\left[\left.{\color[rgb]{0,0,0}-X\frac{e^{-\frac{Y}{\lambda}}}{E_{P}\left[\left.e^{-\frac{Y}{\lambda}}\right|\mathcal{F}_{t}\right]}}\right|\mathcal{F}_{t}\right],$

(see [29] and [32]), solving a BSDE with driver $g_{\Lambda}(s,z,z^{y})=\nabla
g_{\rho}(s,z^{y})\cdot z=\frac{1}{\lambda}z\cdot z^{y}$.

Consider now the following _two alternative drivers_ $g_{\Lambda}$:

$g^{1}_{\Lambda}(s,z,z^{y})=c(z-z^{y})+\frac{1}{2\lambda}\|z^{y}\|^{2};\quad
g^{2}_{\Lambda}(s,z,z^{y})=\frac{1}{2\tilde{\lambda}}\|z-z^{y}\|^{2}+\frac{1}{2\lambda}\|z^{y}\|^{2}$

for some $c,\tilde{\lambda}>0$ where $\tilde{\lambda}$ can be seen as a second
risk aversion parameter. Note that
$g^{1}_{\Lambda}(s,z,z)=g^{2}_{\Lambda}(s,z,z)=g_{\rho}(s,z)$,
$g^{1}_{\Lambda}$ is a Lipschitz driver (in $z$), while $g^{2}_{\Lambda}$ is a
quadratic driver (in $z$) of a similar form of $g_{\rho}$, hence guaranteeing
existence and uniqueness of the solution.

Starting with $g^{1}_{\Lambda}$,

$\displaystyle\Lambda_{t}^{1}(X;Y)-\rho_{t}(Y)$ $\displaystyle=$
$\displaystyle-(X-Y)+\int_{t}^{T}\left[c(Z_{s}^{X,Y}-Z_{s}^{Y})+\frac{1}{2\lambda}\|Z_{s}^{Y}\|^{2}\right]\,ds-\int_{t}^{T}Z_{s}^{X,Y}\,dB_{s}+$
$\displaystyle-\int_{t}^{T}\frac{1}{2\lambda}\|Z_{s}^{Y}\|^{2}\,ds+\int_{t}^{T}Z_{s}^{Y}\,dB_{s}$
$\displaystyle=$
$\displaystyle-(X-Y)+\int_{t}^{T}\left[c(Z_{s}^{X,Y}-Z_{s}^{Y})\right]\,ds-\int_{t}^{T}\left(Z_{s}^{X,Y}-Z_{s}^{Y}\right)\,dB_{s},$

hence

$\displaystyle\Lambda_{t}^{1}(X;Y)$ $\displaystyle=$
$\displaystyle\rho_{t}(Y)-E_{Q_{t,c}}[-(Y-X)|\mathcal{F}_{t}]$
$\displaystyle=$
$\displaystyle\lambda\ln\left(E_{P}\left[\left.\exp\left(-\frac{Y}{\lambda}\right)\right|\mathcal{F}_{t}\right]\right)-E_{Q_{t,c}}[-(Y-X)|\mathcal{F}_{t}],$

with $\frac{dQ_{t,c}}{dP}=\mathcal{E}(c\cdot
B)=\exp\left\\{-\frac{1}{2}c^{2}t-c\,B_{t}\right\\}$.

Concerning $g^{2}_{\Lambda}$, instead,

$\displaystyle\Lambda_{t}^{2}(X;Y)-\rho_{t}(Y)$ $\displaystyle=$
$\displaystyle-(X-Y)+\int_{t}^{T}\left[\frac{1}{2\tilde{\lambda}}\|Z_{s}^{X,Y}-Z_{s}^{Y}\|^{2}+\frac{1}{2\lambda}\|Z_{s}^{Y}\|^{2}\right]\,ds-\int_{t}^{T}Z_{s}^{X,Y}\,dB_{s}+$
$\displaystyle-\int_{t}^{T}\frac{1}{2\lambda}\|Z_{s}^{Y}\|^{2}\,ds+\int_{t}^{T}Z_{s}^{Y}\,dB_{s}$
$\displaystyle=$
$\displaystyle-(X-Y)+\int_{t}^{T}\left[\frac{1}{2\tilde{\lambda}}\|Z_{s}^{X,Y}-Z_{s}^{Y}\|^{2}\right]\,ds-\int_{t}^{T}\left(Z_{s}^{X,Y}-Z_{s}^{Y}\right)\,dB_{s}.$

Since given a constant $a>0$, a final condition $\xi$ and a fixed process
$(b_{t})_{t\in[0,T]}$ the BSDE

$Y_{t}=-\xi+\int_{t}^{T}\frac{1}{2a}\|Z_{s}-b_{s}\|^{2}\,ds-\int_{t}^{T}(Z_{s}-b_{s})\,dB_{s}$

admits a unique solution with first component
$Y_{t}=a\ln\left(E_{P}[\exp(-\frac{\xi}{a})|\mathcal{F}_{t}]\right)$,
$\Lambda_{t}^{2}$ is then given by

$\displaystyle\Lambda_{t}^{2}(X;Y)$ $\displaystyle=$
$\displaystyle\rho_{t}(Y)+\tilde{\lambda}\ln\left(E_{P}\left[\left.\exp\left(-\frac{X-Y}{\tilde{\lambda}}\right)\right|\mathcal{F}_{t}\right]\right)$
$\displaystyle=$
$\displaystyle\ln\left[\left(E_{P}\left[\left.\exp\left(-\frac{Y}{\lambda}\right)\right|\mathcal{F}_{t}\right]\right)^{\lambda}\left(E_{P}\left[\left.\exp\left(-\frac{X-Y}{\tilde{\lambda}}\right)\right|\mathcal{F}_{t}\right]\right)^{\tilde{\lambda}}\right].$

As seen above, we can associate several dynamic CARs to the entropic risk
measure (as well as to any risk measure). In particular, the choice of
$g_{\Lambda}$ and of the corresponding dynamic CAR may reflect the preference
(or the risk aversion) of the agent/intermediary/investor. This financial
interpretation is particularly evident for $g_{\Lambda}^{2}$ depending on the
two risk aversion parameters $\lambda,\tilde{\lambda}$ where $\lambda$ can be
seen as the risk aversion at the level of the whole portfolio $Y$ while
$\tilde{\lambda}$ at the level of the rest of the portfolio with the exclusion
of $X$. In particular, $g_{\Lambda}^{2}$ incorporates the drivers of two
different entropic terms with different risk aversion and, consequently, the
corresponding CAR is somehow related to different entropic risk measures
acting on $Y$ and on $Y-X$.

Note that $g_{\Lambda}^{1},g_{\Lambda}^{2}$ of the previous example are
particular cases of
$g^{f}_{\Lambda}(s,z,z^{y})=f(s,z^{y},z-z^{y})+g_{\rho}(s,z^{y})$ with
$f(s,z^{y},0)=0$. In such a case,

$\displaystyle\Lambda_{t}^{f}(X;Y)$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{T}\left[f(s,Z_{s}^{Y},Z_{s}-Z_{s}^{Y})+g_{\rho}(s,Z_{s}^{Y})\right]\,ds-\int_{t}^{T}Z_{s}\,dB_{s}$
$\displaystyle=$
$\displaystyle-(X-Y)+\int_{t}^{T}f(s,Z_{s}^{Y},Z_{s}-Z_{s}^{Y})\,ds-\int_{t}^{T}\left[Z_{s}-Z_{s}^{Y}\right]\,dB_{s}+$
$\displaystyle-Y+\int_{t}^{T}g_{\rho}(s,Z_{s}^{Y})\,ds-\int_{t}^{T}Z_{s}^{Y}\,dB_{s}$
$\displaystyle=$
$\displaystyle\rho_{t}(Y)-(X-Y)+\int_{t}^{T}(Z_{s}-Z_{s}^{Y})\frac{f(s,Z_{s}^{Y},Z_{s}-Z_{s}^{Y})}{Z_{s}-Z_{s}^{Y}}\,ds-\int_{t}^{T}\left(Z_{s}-Z_{s}^{Y}\right)\,dB_{s}$
$\displaystyle=$
$\displaystyle\rho_{t}(Y)-E_{Q_{t,f}}[-(Y-X)|\mathcal{F}_{t}]$

with $\frac{dQ_{t,f}}{dP}=\mathcal{E}(\frac{\Delta f}{\Delta z}\cdot B)$ and
$\frac{\Delta f}{\Delta
z}=\frac{f(s,Z_{s}^{Y},Z_{s}-Z_{s}^{Y})}{Z_{s}-Z_{s}^{Y}}$. In other words,
when the driver of the CAR is given by that of the risk measure plus an
additional term depending on the difference $z-z^{y}$, the dynamic CAR can be
obtained by the riskiness of the whole portfolio with a correction term
depending on $Y-X$, that is, on the rest of the portfolio with the exclusion
of $X$. Roughly speaking, this approach is somehow similar to that of marginal
contributions where the contribution of $X$ on the whole portfolio $Y$ is
taken into account.

The main aim of this paper has been to provide an axiomatic study of dynamic
CARs and to introduce a general formulation of them in a BSDE-framework, both
under the subdifferentiability assumption, weaker than Gateaux
differentiability. The following remark illustrates why the
subdifferentiability assumption is rather weak for $g_{\rho}$ and what is the
impact of subdiffertiability of $g_{\Lambda}$ on $\Lambda_{t}$ and on
$\rho_{t}$.

###### Remark 13

Although $g_{\rho}$ is assumed to be convex in $z\in\mathbb{R}^{n}$ (because
of convexity of $\rho_{t}$), $g_{\Lambda}(s,z,z^{y})$ is not necessarily
convex in $z$. Consequently, while $\partial g_{\rho}(t,z)\neq\emptyset$ on
the relative interior of the domain of $g_{\rho}$ (see Rockafellar [38]),
$g_{\Lambda}(s,z,z^{y})$ is not necessarily subdifferentiable in $z$.

We claim that subdifferentiability of $g_{\Lambda}$ in $z$, however, implies
subdifferentiability of $g_{\rho}$, $\rho_{t}$ and $\Lambda_{t}$, where that
of $\Lambda_{t}$ should be understood as

$\partial\Lambda_{t}(X;Y)=\left\\{Q_{t}\in\mathcal{Q}_{t}:\Lambda_{t}(H;Y)\geq\right.\\\
\left.\Lambda_{t}(X;Y)+E_{Q_{t}}[-(H-X)|\mathcal{F}_{t}]\mbox{ for all }H\in
L^{\infty}(\mathcal{F}_{T})\right\\}\neq\emptyset$

for any $Y\in L^{\infty}(\mathcal{F}_{T})$.

In fact, subdifferentiability of $g_{\rho}$ follows immediately since
$g_{\rho}(s,z)=g_{\Lambda}(s,z,z)$.

Subdifferentiability of $g_{\Lambda}$, then, implies that

$\displaystyle\Lambda_{t}(H;Y)-\Lambda_{t}(X;Y)$ $\displaystyle=$
$\displaystyle-(H-X)+\int_{t}^{T}\left[g_{\lambda}(s,Z_{s}^{H,Y},Z_{s}^{Y})-g_{\lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})\right]\,ds-\int_{t}^{T}\left[Z^{H,Y}_{s}-Z_{s}^{X,Y}\right]\,dB_{s}$
$\displaystyle\geq$ $\displaystyle-(H-X)+\int_{t}^{T}\partial
g_{\lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})\left[Z_{s}^{H,Y}-Z_{s}^{X,Y}\right]\,ds-\int_{t}^{T}\left[Z^{H,Y}_{s}-Z_{s}^{X,Y}\right]\,dB_{s}$
$\displaystyle=$ $\displaystyle E_{Q_{t}^{Y,X}}[-(H-X)|\mathcal{F}_{t}],$

for any $H,X,Y\in L^{\infty}(\mathcal{F}_{T})$, where
$E\left[\left.\frac{dQ_{t}^{Y,X}}{dP}\right|\mathcal{F}_{t}\right]=\mathcal{E}(\partial
g_{\lambda}(s,Z_{s}^{X,Y},Z_{s}^{Y})\cdot B)$. It then follows that
$Q_{t}^{Y,X}\in\partial\Lambda_{t}(X;Y)$, so both $\Lambda_{t}(X;Y)$ and
$\rho_{t}(Y)$ are subdifferentiable.

##  5 Conclusions

In this paper, we have introduced a general axiomatic approach to dynamic
capital allocations as well as an approach suitable for risk measures induced
by $g$-expectations, by going beyond the gradient approach and by weakening
the Gateaux differentiability condition.

Furthermore, for risk measures induced by $g$-expectations we have seen that
it is possible to associate several dynamic capital allocations induced by a
$g_{\Lambda}$-expectation, where the choice of the driver $g_{\Lambda}$ and of
the corresponding dynamic CAR may reflect the preferences of the
intermediary/investor. Vice versa, instead, given a dynamic family of CARs
induced by a $g$-expectation, the associated dynamic risk measure is uniquely
determined.

Acknowledgments. We wish to thank two anonymous Referees for their comments
that contributed to improve the paper. We are also grateful to Alessandro
Doldi, Marco Frittelli, and Marco Zullino for their suggestions and
discussions.

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