# Capital allocation for cash-subadditive risk measures: from BSDEs to BSVIEs

Emanuela Rosazza Gianin111Department of Statistics and Quantitative Methods,
University of Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano,
Italy<EMAIL_ADDRESS>and Marco Zullino 222Department of
Mathematics and Applications, University of Milano Bicocca, 20126 Milano,
Italy<EMAIL_ADDRESS>
###### Abstract

In the context of risk measures, the capital allocation problem is widely
studied in the literature where different approaches have been developed, also
in connection with cooperative game theory and systemic risk. Although static
capital allocation rules have been extensively studied in the recent years,
only few works deal with dynamic capital allocations and its relation with
BSDEs. Moreover, all those works only examine the case of an underneath risk
measure satisfying cash-additivity and, moreover, a large part of them focuses
on the specific case of the gradient allocation where Gateaux
differentiability is assumed.

The main goal of this paper is, instead, to study general dynamic capital
allocations associated to cash-subadditive risk measures, generalizing the
approaches already existing in the literature and motivated by the presence of
(ambiguity on) interest rates. Starting from an axiomatic approach, we then
focus on the case where the underlying risk measures are induced by BSDEs
whose drivers depend also on the $y$-variable. In this setting, we
surprisingly find that the corresponding capital allocation rules solve
special kinds of Backward Stochastic Volterra Integral Equations (BSVIEs).

Keywords: risk measures; capital allocation; BSDE; BSVIE; cash-subadditivity;
subdifferential

##  1 Introduction

Starting from the seminal paper of Artzner et al. [3], a wide literature on
Mathematical Finance has been devoted to the theory of risk measures that have
been introduced and studied - both from an axiomatic and an empirical point of
view- in order to quantify the riskiness of financial exposures in a static
and in a dynamic setting. See [3, 6, 13, 18, 22, 24, 5, 11, 23, 16, 20], among
many others, for an axiomatic treatment of the topic. In the context of
dynamic risk measures, the relation between risk measures and backward
equations have been deeply analysed: firstly, for Backward Stochastic
Differential Equations (BSDEs) in [4, 14, 17, 20, 38]; and, recently, for
Backward Stochastic Volterra Integral Equations (BSVIEs) in [1, 17, 45]. We
recall that, while BSDEs date back to [34, 35] and their applications to
Mathematical Finance are well known in the literature (see [19]), BSVIEs were
introduced for the first time by [44] and recently investigated also in view
of their applications.

During the last years and in the context of risk measures, an increasing
number of studies has been focused on the capital allocation problem that,
roughly speaking, consists of finding a “fair” way (i.e. satisfying some
suitable criteria) to divide into the different components of the risk the
margin to be deposited because of the risk exposure. To be more concrete, with
different components we can think at different sub-units which a risky
position is formed of (e.g. a portfolio composed by different assets or a firm
formed by different business lines or branches). On the capital allocation
problem, several papers deal with an axiomatic approach in a static framework
and with its relation with cooperative game theory (see [10, 13, 15, 26, 39,
41] and the references therein). Some recent works, instead, focus also on
capital allocation rules (CARs) in the dynamic case, even if several of those
restrict their attention mainly on the gradient allocation (see, e.g., [7, 28,
30, 40]). Furthermore, quite recently, the strong relation between risk
measures and BSDEs led to a natural extension of the concept of a capital
allocation rule to a dynamic environment by means of BSDEs. In particular,
[28] has proved that the dynamic gradient capital allocation of a dynamic risk
measure induced by a BSDE follows a BSDE. Moreover, [33] gives an organic and
axiomatic treatment of general dynamic CARs as well as a general construction
of dynamic CARs by means of BSDEs.

It is worth emphasizing that all the aforementioned papers deal with capital
allocation rules associated to (coherent or convex) cash-additive risk
measures. As argued by [20], however, in the presence of stochastic interest
rate and/or ambiguity on the interest rate the axiom of cash-additivity should
be relaxed, e.g., with the so-called cash-subadditivity. See also [11, 18, 23,
31, 32] for the impact of non cash-additivity on risk measures and for a
deeper discussion on cash-subadditivity versus cash-additivity.

Motivated by the importance of studying general dynamic CARs and by the
financial need of weakening cash-subadditivity, our aim is here to develop an
axiomatic treatment of CARs based on dynamic risk measures that satisfy cash-
subadditivity but not necessarily Gateaux differentiability. In order to
pursue this objective, we generalize the approach followed in [33], covering
the case of cash-subadditivity. Differently from [31, 30, 28] where the
authors proved that for cash-additive risk measures induced by BSDEs (or
BSVIEs) the corresponding capital allocations follow still a BSDE (or a
BSVIE), in this paper we prove that the subdifferential CAR associated to a
cash-subadditive risk measure induced via a BSDE follows a BSVIE. More in
general, we introduce a general technique to build capital allocations
following BSVIEs, similarly as done for cash-additive risk measures in [33,
31], and we provide a link between dynamic cash-subadditive risk measures
following a BSDEs and capital allocations based on BSVIEs. This new approach
will be shown to cover relevant examples of capital allocation principles,
such as the gradient, the Aumann-Shapley, and the marginal ones.

The paper is organized as follows. In Section 2 we give a short review on
dynamic risk measures, capital allocation rules, BSDEs, and BSVIEs. Section 3
contains our main results. Firstly, we provide a one-to-one correspondence
between dynamic cash-subadditive convex risk measures and capital allocations
rules satisfying some further properties, by generalizing a similar result
proved in [33] for the cash-additive case. Furthermore, focusing on CARs whose
underlying cash-subadditive risk measures are driven by BSDEs, we show that
the subdifferential CAR follows a BSVIE. More in general, we provide a natural
way to build different CARs whose dynamics follow a BSVIE with a driver that
is related to that of the associated risk measure. Section 4 contains some
examples based on marginal capital allocations and on the entropic risk
measure. Finally, in Section 5 some conclusions and final remarks are
provided.

##  2 Short review and preliminary results

In this section we fix some notations and we recall the notions that will be
used in the following. We start fixing some relevant notations used throughout
the paper, then we recall fundamental properties of dynamic risk measures.
Furthermore, we provide an insight into the main results regarding BSDEs and
BSVIEs, proving some new and useful results for our specific purposes.

Let $T>0$ be a future time horizon and consider a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ with
$\mathcal{F}_{0}=\\{\emptyset,\Omega\\}$ and $\mathcal{F}_{T}=\mathcal{F}$. In
the sequel, any equality and inequality has to be understood
$\mathbb{P}$-almost surely (shortly, a.s.), unless otherwise specified.

Let $L^{p}(\mathcal{F}_{t})$, with $p\in[1,+\infty)$, denote the space formed
by all ($\mathbb{R}^{n}$-valued) random variables on
$(\Omega,\mathcal{F},\mathbb{P})$ that are $\mathcal{F}_{t}$-measurable and
such that $\mathbb{E}\left[|X|^{p}\right]<+\infty$, while
$L^{\infty}(\mathcal{F}_{t})$ the space formed by all
($\mathbb{R}^{n}$-valued) random variables that are
$\mathcal{F}_{t}$-measurable and essentially bounded. Furthermore,
$L^{p}_{+}(\mathcal{F}_{t})$, with $p\in[1,+\infty]$, will denote the subset
of $L^{p}(\mathcal{F}_{t})$ of random variables that are, $\mathbb{P}$-a.s.,
greater than or equal to $0$.

###  2.1 Dynamic risk measures

We recall that static risk measures have been introduced to quantify now the
riskiness of financial instruments (or, more precisely, of their profits and
losses or returns) “expiring” at a given time horizon $T$, while dynamic risk
measures are developed to quantify such riskiness at any time between now and
the maturity $T$. To be more precise, a static risk measure on $L^{\infty}$ is
a functional $\rho:L^{\infty}(\mathcal{F}_{T})\to\mathbb{R}$ satisfying some
further financially reasonable axioms, while a dynamic risk measure is a
family $(\rho_{t})_{t\in[0,T]}$, with $\rho_{t}:L^{\infty}(\mathcal{F}_{T})\to
L^{\infty}(\mathcal{F}_{t})$, such that $\rho_{0}$ is a static risk measure
and $\rho_{T}(X)=-X$ for any $X\in L^{\infty}(\mathcal{F}_{T})$. Note that any
$X\in L^{\infty}(\mathcal{F}_{T})$ has to be understood as the profit and loss
of a financial position (where positive values stand for profits, while
negative for losses).

For a comprehensive literature on risk measures, we address the reader to [3,
13, 22, 24, 6, 4, 11, 23, 16, 20, 38], among many others.

Here below we collect a non exhaustive list of desirable axioms that may be
assumed for (dynamic) risk measures:

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

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

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

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

\- 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
$X\in L^{\infty}(\mathcal{F}_{T}),0\leq s\leq t\leq T$.

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

\- continuity from below (resp. above): for any sequence
$(X_{n})_{n\in\mathbb{N}}\subseteq L^{\infty}(\mathcal{F}_{T})$ such that
$X_{n}\uparrow X$ a.s. (resp. $X_{n}\downarrow X$ a.s.) with $X\in
L^{\infty}(\mathcal{F}_{T})$, it holds that
$\rho_{t}(X_{n})\downarrow\rho_{t}(X)$ a.s. (resp.
$\rho_{t}(X_{n})\uparrow\rho_{t}(X)$ a.s.).

In particular, a dynamic risk measure will be said to be convex if it
satisfies monotonicity, convexity and normalization.

(Decreasing) Monotonicity, convexity, normalization, and continuity from
below/above are quite commonly assumed for risk measures. In particular,
convexity is related to diversification of risk, while monotonicity to the
interpretation of risk measures where the riskier is the financial position
$X$, the greater is the value of $\rho(X)$. The riskiness of having zero
profit and loss is set to be $0$ when normalization holds. Cash-additivity,
instead, guarantees that, if we add some cash (or some amount that is known at
time $t$) to the financial position $X$, then the riskiness of the position is
diminished exactly of the cash amount. Somehow, then, cash-additivity assumes
zero interest rate and provides the financial interpretation of risk measures
in terms of capital requirements (or margins). As argued in [20], however, in
presence of stochastic interest rate or ambiguity on interest rates it should
be financially reasonable to replace cash-additivity with cash-subadditivity.

Finally, time-consistency and weak time-consistency provide the inter-temporal
behavior of risk measures. While time-consistency means that at any time $s$
it is indifferent to evaluate the riskiness of a financial position $X$ in one
step or in two (or more) steps backwards, weak time-consistency assumes that
the riskiness measured in one step is lower than that evaluated in more steps.

We recall from [5] that monotonicity together with cash-additivity implies

\- regularity:
$\rho_{t}(X\mathbb{I}_{A}+Y\mathbb{I}_{A^{C}})=\rho_{t}(X)\mathbb{I}_{A}+\rho_{t}(Y)\mathbb{I}_{A^{C}}$
for any $t\in[0,T],A\in\mathcal{F}_{t},X,Y\in L^{\infty}(\mathcal{F}_{T})$.

It can be proved that also cash-subadditivity and monotonicity are enough to
ensure regularity.

Concerning dual representations of risk measures, it is well known that:

\- any dynamic convex risk measure on $L^{\infty}(\mathcal{F}_{T})$ satisfying
cash-additivity and continuity from below has the following dual
representation (see [16]):

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

where, for any $t\in[0,T]$,

$\mathcal{Q}\triangleq\\{\mathbb{Q}_{t}\mbox{ probability measures on
}(\Omega,\mathcal{F}):\mathbb{Q}_{t}\ll\mathbb{P}\mbox{ and
}\mathbb{Q}_{t}|_{\mathcal{F}_{t}}=\mathbb{P}|_{\mathcal{F}_{t}}\\}$

and $c_{t}$ is a non-negative $\mathcal{F}_{t}$-measurable penalty function;

\- any dynamic convex risk measure $L^{\infty}(\mathcal{F}_{T})$ satisfying
cash-subadditivity and continuity from above has the following dual
representation (see [33]):

$\rho_{t}(X)=\operatorname*{ess\,sup}_{(D_{t},\mathbb{Q}_{t})\in\mathcal{D}\times\mathcal{Q}}\\{D_{t}\mathbb{E}_{\mathbb{Q}_{t}}[-X|\mathcal{F}_{t}]-c_{t}(D_{t}\mathbb{Q}_{t})\\},$
(2)

where

$\mathcal{D}\triangleq\\{(D_{t})_{t\in[0,T]}\mbox{ adapted processes w.r.t.
}({\mathcal{F}_{t}})_{t\in[0,T]}:D_{t}\in[0,1]\,\mbox{ for every
}t\in[0,T],\mbox{ a.s.}\\}$

can be interpreted as a set of discounting factors, while
$c_{t}(D_{t}\mathbb{Q}_{t})$ is a non-negative $\mathcal{F}_{t}$-measurable
random variable which plays the role of a generalized penalty function.

Henceforth, we will always impose that the essential supremum in (2) is
attained at some
$(\bar{D}_{t},\bar{\mathbb{Q}}_{t})\in\mathcal{D}\times\mathcal{Q}$, i.e.

$(\bar{D}_{t},\bar{\mathbb{Q}}_{t})\in\operatorname*{arg\,ess\,max}_{(D_{t},\mathbb{Q}_{t})\in\mathcal{D}\times\mathcal{Q}}\,\\{D_{t}\mathbb{E}_{\mathbb{Q}_{t}}[-X|\mathcal{F}_{t}]-c_{t}(D_{t}\mathbb{Q}_{t})\\}.$

In such a case, $(\bar{D}_{t},\bar{\mathbb{Q}}_{t})$ is referred to as an
optimal scenario and

$\rho_{t}(X)=\bar{D}_{t}\mathbb{E}_{\bar{\mathbb{Q}}_{t}}[-X|\mathcal{F}_{t}]-c_{t}(\bar{D}_{t}\bar{\mathbb{Q}}_{t}).$
(3)

The following result guarantees that this holds true for cash-subadditive
convex risk measures satisfying continuity from below, similarly as for the
cash-additive case.

######  Proposition 1.

If $\rho_{t}:L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$ is a
dynamic convex cash-subadditive risk measure satisfying continuity from below,
then

$\rho_{t}(X)=\operatorname*{ess\,max}_{(D_{t},\mathbb{Q}_{t})\in\mathcal{D}\times\mathcal{Q}}\,\left\\{\mathbb{E}_{\mathbb{Q}_{t}}\left[-D_{t}X|\mathcal{F}_{t}\right]-c_{t}(D_{t}\mathbb{Q}_{t})\right\\}=\mathbb{E}_{\bar{\mathbb{Q}}_{t}}\left[-\bar{D}_{t}X|\mathcal{F}_{t}\right]-c_{t}(\bar{D}_{t}\bar{\mathbb{Q}}_{t})$
(4)

for some $(\bar{D}_{t},\bar{\mathbb{Q}}_{t})\in\mathcal{D}\times\mathcal{Q},$
where $c_{t}$ is the minimal penalty term, i.e.

$c_{t}(D_{t}\mathbb{Q}_{t})\triangleq\operatorname*{ess\,sup}_{X\in
L^{\infty}(\mathcal{F}_{T})}\left\\{D_{t}\mathbb{E}_{\mathbb{Q}_{t}}[-X|\mathcal{F}_{t}]-\rho_{t}(X)\right\\}.$
(5)

###### Proof.

Let $\rho_{t}$ be a dynamic convex, cash-subadditive, and continuous from
below risk measure. It is then easy to check that
$\rho_{0,t}=\mathbb{E}[\rho_{t}(X)]$ is a static risk measure with the same
properties. According to the proof of Theorem 4.3 in [20] and to the
representation of convex static risk measures that are continuous from below
(see [4] or [22, 23] for a thorough discussion), it follows that

$\rho_{0,t}(X)=\max_{\mu\in\mathcal{S}}\left\\{\mathbb{E}_{\mu}(-X)-c_{0,t}(\mu)\right\\}=\mathbb{E}_{\bar{\mu}}(-X)-c_{0,t}(\bar{\mu}),$
(6)

where $c_{0,t}$ is the minimal penalty function of $\rho_{0,t}$ and
$\mathcal{S}$ is the set formed by all sub-probabilities $\mu$ on
$(\Omega,\mathcal{F})$ that are absolutely continuous w.r.t. $\mathbb{P}$. We
recall (see [20] for further details) that with sub-probability it is meant a
measure $\mu:(\Omega,\mathcal{F})\to[0,1]$ such that $0\leq\mu(\Omega)\leq 1$.

From Proposition 6 in [31], the sub-probability $\bar{\mu}$ can be decomposed
as $\bar{\mu}=\bar{D}_{t}\bar{\mathbb{Q}}_{t}$ with
$(\bar{D}_{t},\bar{\mathbb{Q}}_{t})\in\mathcal{D}\times\mathcal{Q}$ whenever
$c_{0,t}(\bar{\mu})<+\infty$; moreover,
$c_{0,t}(\bar{D}_{t}\bar{\mathbb{Q}}_{t})=\mathbb{E}[c_{t}(\bar{D}_{t}\bar{\mathbb{Q}}_{t})]$.
Hence

$\rho_{0,t}(X)=\mathbb{E}[\rho_{t}(X)]=\mathbb{E}[{\bar{D}_{t}\mathbb{E}_{\bar{\mathbb{Q}}_{t}}}[-X|\mathcal{F}_{t}]-c_{t}(\bar{D}_{t}\bar{\mathbb{Q}}_{t})].$
(7)

The arguments above imply that

$\rho_{t}(X)=\operatorname*{ess\,sup}_{(D_{t},\mathbb{Q}_{t})\in\mathcal{D}\times\mathcal{Q}}\left\\{D_{t}\mathbb{E}_{\mathbb{Q}_{t}}{[-X|\mathcal{F}_{t}]-c_{t}(D_{t}\mathbb{Q}_{t})}\right\\}\geq\bar{D}_{t}\mathbb{E}_{\bar{\mathbb{Q}}_{t}}[-X|\mathcal{F}_{t}]-c_{t}(\bar{D}_{t}\bar{\mathbb{Q}}_{t}),$

hence

$\rho_{t}(X)-\Big{(}\bar{D}_{t}\mathbb{E}_{\bar{\mathbb{Q}}_{t}}[-X|\mathcal{F}_{t}]-c_{t}(\bar{D}_{t}\bar{\mathbb{Q}}_{t})\Big{)}\geq
0$

and, by (7),

$\mathbb{E}\left[\rho_{t}(X)-\Big{(}\bar{D}_{t}\mathbb{E}_{\bar{\mathbb{Q}}_{t}}[-X|\mathcal{F}_{t}]-c_{t}(\bar{D}_{t}\bar{\mathbb{Q}}_{t})\Big{)}\right]=0.$

The thesis then follows. ∎

In full generality, we say that a cash-subadditive convex risk measure is
representable if

$\rho_{t}(X)=\operatorname*{ess\,sup}_{(D_{t},\mathbb{Q}_{t})\in\mathcal{D}^{\prime}\times\mathcal{Q}}\left\\{\mathbb{E}_{\mathbb{Q}_{t}}[-D_{t}X|\mathcal{F}_{t}]-c_{t}(D_{t}\mathbb{Q}_{t})\right\\}$
(8)

for some $D_{t}\in\mathcal{D}^{\prime}$, where $\mathcal{D}^{\prime}$ is
formed by all (not necessarily adapted) stochastic processes
$(D_{t})_{t\in[0,T]}$ that are valued in $[0,1]$.

###  2.2 Capital allocation rules

In this section, we briefly recall the capital allocation problem, in a static
and dynamic setting, and with an axiomatic approach (see, e.g., [7, 9, 12, 15,
26, 29, 39, 40]). The first axiomatic approach is given by Kalkbrener [26] is
a static setting, while the dynamic setting has been firstly studied in [28]
for risk measures induced via BSDEs and in the case of gradient allocation.
Nevertheless, the first organic work about (general) dynamic capital
allocations is provided in Mastrogiacomo and Rosazza Gianin [33] where the
approach of [26] is generalized to a dynamic setting.

According to the classical literature, a static capital allocation rule
consists of deciding how to divide the capital requirement (or margin) of an
aggregate risky position $X$ into the sub-portfolios (also called business
lines) which $X$ is formed of, according to some financially sound criteria.
To be more concrete, given a static risk measure $\rho$, an aggregate position
$X$ and its business lines $X_{1},...,X_{n}$ (hence such that
$X=X_{1}+...+X_{n}$), a capital allocation rule prescribes how to share
$\rho(X)$ into $X_{1},...,X_{n}$ by allocating the capital $k_{i}$ to $X_{i}$
such that $\rho(X)=\sum_{i=1}^{n}k_{i}$ (when the so called full allocation of
the capital allocation principle is fulfilled).

Starting from those considerations, Kalkbrener [26] introduced the notion of
capital allocation rule (CAR, for short) associated to a risk measure $\rho$
as a functional $\Lambda:L^{\infty}(\mathcal{F}_{T})\times
L^{\infty}(\mathcal{F}_{T})\to\mathbb{R}$ such that $\Lambda(X,X)=\rho(X)$ for
any $X\in L^{\infty}$. In other words, the capital to be allocated to the
whole aggregate portfolio $X$ (as a stand-alone portfolio) coincides with the
riskiness of $X$ evaluated via the static risk measure $\rho$. Furthermore,
$\Lambda(Y,X)$ should be understood as the capital to be allocated to $Y$ as a
sub-portfolio of $X$.

We now recall the definition of a dynamic CAR.

######  Definition 2 (see [33]).

Let $(\rho_{t})_{t\in[0,T]}$ be a dynamic risk measure.

A family of functionals $\Lambda_{t}:L^{\infty}(\mathcal{F}_{T})\times
L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$, with $t\in[0,T]$,
is said to be a dynamic CAR associated to the risk measure
${(\rho_{t})}_{t\in[0,T]}$ if

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

Instead, the family ${(\Lambda_{t})}_{t\in[0,T]}$ is called audacious CAR if
$\Lambda_{t}(X,X)\leq\rho_{t}(X)$, for any $t\in[0,T]$ and $X\in
L^{\infty}(\mathcal{F}_{T})$.

In other words, while a dynamic CAR requires that the capital to be allocated
to $X$ as a stand-alone portfolio should be equal to the margin $\rho_{t}(X)$,
an audacious one only asks to allocate not more than the margin (see [10, 33]
and the references therein for a discussion):

In the sequel, when no confusion may occur, we identify the family of
functionals ${(\Lambda_{t})}_{t\in[0,T]}$ with an element $\Lambda_{t}$. Here
below, we provide a (non exhaustive) list of axioms that could be sometimes
imposed to a dynamic CAR. See [33] and the references therein for more
details.

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

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

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

\- cash-subadditivity:
$\Lambda_{t}(X+m_{t},Y+m_{t})\geq\Lambda_{t}(X,Y)-m_{t}$ for any $t\in[0,T]$,
$m_{t}\in L_{+}^{\infty}(\mathcal{F}_{t}),X,Y\in L^{\infty}(\mathcal{F}_{T})$.

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

\- 1-cash-subadditivity: $\Lambda_{t}(X+m_{t},Y)\geq\Lambda_{t}(X,Y)-m_{t}$
for any $t\in[0,T]$, $m_{t}\in L^{\infty}_{+}(\mathcal{F}_{t}),X,Y\in
L^{\infty}(\mathcal{F}_{T})$.

\- normalization: $\Lambda_{t}(0,X)=0$ for any $t\in[0,T],X\in
L^{\infty}(\mathcal{F}_{T})$.

\- full allocation:
$\sum_{i=1}^{n}\Lambda_{t}\left(X_{i},X\right)=\Lambda_{t}(X,X)$ for any
$t\in[0,T]$ and $(X_{i})_{i=1,\dots,n},X\in L^{\infty}(\mathcal{F}_{T})$ with
$\sum_{i=1}^{n}X_{i}=X$.

We shortly recall the financial interpretation and motivation of the previous
axioms (see, for instance, [15, 10, 26, 33, 39, 40] for a deeper discussion).
Firstly, monotonicity means that the riskier a position is, the greater is the
capital allocated needed to cover its risk. No-undercut property guarantees
that the risk of any sub-portfolio $X$ of $Y$ cannot exceed the total risk of
portfolio $X$ seen as a stand-alone portfolio, implying therefore that there
is no incentive to split $X$ from $Y$ (in the terminology of Tsanakas [41]).
Cash-additivity conveys the principle according to which any addition of
’known’ (or, better, $\mathcal{F}_{t}$-measurable) risk $m_{t}$ to both the
sub-unit $X$ and the position $Y$ yields a decrement of the risk exactly equal
to $m_{t}$. Cash-subadditivity substitutes cash-additivity in the case of the
presence of ambiguous rate interest in the market (in the same spirit of El
Karoui and Ravanelli [20] for risk measures). Similar interpretations hold for
1-cash-additivity and 1-cash-subadditivity. Normalization requires that the
risk associated to an initial null sub-portfolio remains equal to zero at
every time $t\in[0,T]$. The full allocation requirement seems to be quite
natural in the context of capital allocation rules because it implies that the
total contribution to the risk of the sum of every sub-portfolio is exactly
equal to the risk of the overall position. However, as from [26], full
allocation and no-undercut properties could be incompatible for capital
allocations rules when the underlying risk measure is strictly convex and not
coherent. Nevertheless, when a firm is only interested in monitoring its
financial activities, full allocation can be replaced by one of the following
axioms:

\- sub-allocation: for any $t\in[0,T]$,

$\Lambda_{t}\left(X,X\right)\geq\sum_{i=1}^{n}\Lambda_{t}\left(X_{i},X\right)$

holds for any $(X_{i})_{i=1,\dots,n},X\in L^{\infty}(\mathcal{F}_{T})$ with
$\sum_{i=1}^{n}X_{i}=X$.

\- weak-convexity: for any $t\in[0,T]$,

$\Lambda_{t}\left(\displaystyle\sum_{i=1}^{n}a_{i}X_{i},X\right)\leq\displaystyle\sum_{i=1}^{n}a_{i}\Lambda_{t}(X_{i},X)$

for any $X,X_{i}\in L^{\infty}(\mathcal{F}_{T}),a_{i}\in[0,1]$, for
$i=1,\dots,n$, with $\sum_{i=1}^{n}a_{i}=1$ and $\sum_{i=1}^{n}a_{i}X_{i}=X$.

In particular, for sub-allocation, the difference between the capital
allocated to cover the risk of all sub-portfolios and the capital stored to
insure the risk of the full portfolio can be understood as an extra-security
margin justified, e.g., by some extra-costs shared among all the sub-
portfolios (see also the discussion in [8, 10, 9]). Weak-convexity, instead,
reflects the diversification of risk that encourages the sharing of risk
between different assets.

Finally, as for risk measures, also for dynamic capital allocations rules it
is relevant to investigate time-consistency. The following two formulations of
time-consistency have been introduced in [33]:

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

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

Such formulations guarantee that, roughly speaking, it is indifferent to
compute the capital allocation with a one-step procedure or proceeding
backwards in time with multiple steps (see [33] for a detailed discussion).

###  2.3 BSDEs and BSVIEs

We shortly review now the main definitions and results on Backward Stochastic
Differential Equations (BSDEs) and Backward Stochastic Volterra Integral
Equations (BSVIEs) used in the sequel. On the probability space
$(\Omega,\mathcal{F},\mathbb{P})$, consider a $n$-dimensional Brownian motion
$(B_{t})_{t\in[0,T]}$ with its augmented natural filtration
$(\mathcal{F}_{t})_{t\in[0,T]}$.

It is well known (see [37]) that, in a Brownian setting, for any probability
measure $\mathbb{Q}\ll\mathbb{P}$ there exists a predictable process
$(q_{t})_{t\in[0,T]}$ such that:

$\mathbb{E}\left[\frac{d\mathbb{Q}}{d\mathbb{P}}\Bigg{|}\mathcal{F}_{t}\right]=\mathcal{E}(q\cdot
W)_{t}\triangleq\exp\left(-\frac{1}{2}\int_{0}^{t}|q_{s}|^{2}ds-\int_{0}^{t}q_{s}dB_{s}\right),$

where $|\cdot|$ stands for the Euclidean norm both in $\mathbb{R}$ and in
$\mathbb{R}^{n}$. Henceforth, any probability measure
$\mathbb{Q}\ll\mathbb{P}$ will be identified with its Radon-Nikodym density
$\frac{d\mathbb{Q}}{d\mathbb{P}}$.

We fix now some notations we will use in the following. Let $p\in[1,+\infty)$,
$0\leq\tau<T$ and $\Delta_{\tau}\triangleq\\{(t,s)\in[\tau,T]^{2}:t\leq s\\}$,
with the convention $\Delta_{0}\triangleq\Delta$. Let us denote

$\displaystyle
L^{p}(\Omega,\mathcal{F}_{t};L^{q}(\tau,T))\triangleq\left\\{\phi:\Omega\times[\tau,T]\to\mathbb{R}:\phi\mbox{
is }\mathcal{F}_{t}\otimes\mathcal{B}([\tau,T])\mbox{-meas. and
}\mathbb{E}\left(\int_{\tau}^{T}|\phi_{s}|^{q}ds\right)^{\frac{p}{q}}<+\infty\right\\},$
$\displaystyle
L^{p}(\Omega,\mathbb{F};L^{q}(\tau,T))\triangleq\left\\{\phi:\Omega\times[\tau,T]\to\mathbb{R}:\phi\mbox{
is }(\mathcal{F}_{t})_{t\in[\tau,T]}\mbox{-adapted and
}\mathbb{E}\left(\int_{\tau}^{T}|\phi_{s}|^{q}ds\right)^{\frac{p}{q}}<+\infty\right\\},$

while let $L^{q}((\tau,T);L^{p}(\Omega,\mathbb{F};L^{m}(\tau,T)))$ denote the
space of all stochastic processes
$\phi:\Omega\times\Delta_{\tau}\to\mathbb{R}$ such that, for any
$t\in[\tau,T]$,

$\phi(t,\cdot)\mbox{ is }(\mathcal{F}_{m})_{m\in[t,T]}-\mbox{adapted and
}\int_{\tau}^{T}\Big{\\{}\mathbb{E}\Big{(}\int_{t}^{T}|\phi(t,s)|^{m}ds\Big{)}^{p/m}\Big{\\}}^{q/p}dt<+\infty.$

Furthermore, let us denote

$\displaystyle\mathcal{H}^{p,q}(\tau,T)\triangleq
L^{p}(\Omega,\mathbb{F};L^{q}(\tau,T))\times
L^{q}((\tau,T),\mathcal{F}_{t};L^{p}(\Omega,\mathbb{F};L^{2}(\tau,T))),$
$\displaystyle\mbox{BMO}({\mathbb{P}})\triangleq\left\\{\phi\in
L^{2}(\Omega,\mathbb{F};L^{2}(\tau,T)):\ \exists C>0\mbox{ s.t., }\forall
t\in[\tau,T],\
\mathbb{E}\left[\left.\int_{t}^{T}|\phi_{s}|^{2}ds\right|\mathcal{F}_{t}\right]\leq
C\mbox{ a.s.}\right\\}$ $\displaystyle
L^{p}((0,T);\mbox{BMO}({\mathbb{P}}))\triangleq\left\\{\phi:\Omega\times\Delta_{\tau}\to\mathbb{R}:\int_{\tau}^{T}\left\\{\left\|\sup_{r\in[t,T]}\mathbb{E}\left[\left.\int_{r}^{T}|\phi(t,s)|^{2}ds\right|\mathcal{F}_{r}\right]\right\|_{\infty}\right\\}^{p}dt<+\infty\right\\}.$

We can also include the case $p=+\infty$, adapting our definitions in the
usual way. To simplify the notation, for $p=q$ we shorten
$L^{p}(\Omega,\mathcal{F}_{t};L^{p}(\tau,T))\triangleq
L^{p}((\tau,T);\mathcal{F}_{t})$ and analogously for the other cases. We
observe that if $\phi\in\mbox{BMO}(\mathbb{P})$ then

$\left\|\sup_{t\in[\tau,T]}\mathbb{E}\left[\left.\int_{t}^{T}|\phi_{s}|^{2}ds\right|\mathcal{F}_{t}\right]\right\|_{\infty}<+\infty.$

####  2.3.1 BSDEs and related cash-subaddditive risk measures

Following [19, 27, 34, 35] and the references therein, a Backward Stochastic
Differential Equation (BSDE) is an equation of the following form

$Y_{t}=X+\int_{t}^{T}g(s,Y_{s},Z_{s})ds-\int_{t}^{T}Z_{s}dB_{s},$ (9)

where $X\in L^{2}(\mathcal{F}_{T})$ is the final condition,
$g:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ is
called driver of the BSDE, while $(Y_{t},Z_{t})_{t\in[0,T]}$ are the unknown
processes where $Y$ is $\mathbb{R}$-valued and $Z$ is $\mathbb{R}^{n}$-valued.
For simplicity, the dependence on $\omega\in\Omega$ is conventionally often
omitted.

We recall the following existence and uniqueness result.

######  Theorem 3 (see [34, 19]).

Let $X\in L^{2}(\mathcal{F}_{T})$ and suppose that
$g:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$
satisfies the following assumptions:

  * a)

Uniformly Lipschitz: there exists $C>0$ such that $d\mathbb{P}\otimes dt$-a.s.

$|g(\omega,t,y_{1},z_{1})-g(\omega,t,y_{2},z_{2})|\leq
C(|y_{1}-y_{2}|+|z_{1}-z_{2}|)\quad\mbox{ for any
}(y_{1},z_{1}),(y_{2},z_{2})\in\mathbb{R}\times\mathbb{R}^{n};$

  * b)

$g$ is
$\mathcal{F}_{T}\times\mathcal{B}([0,T]\times\mathbb{R}\times\mathbb{R}^{n})$-measurable
and $g(t,y,z)$ is an $(\mathcal{F}_{t})_{t\in[0,T]}$-progressively measurable
stochastic process for any $(y,z)\in\mathbb{R}\times\mathbb{R}^{n}$;

  * c)

$g(\cdot,0,0)\in L^{2}((0,T),\mathbb{F})$.

Then there exists a unique pair of stochastic processes
$(Y_{t},Z_{t})_{t\in[0,T]}\in L^{2}((0,T),\mathbb{F})\times
L^{2}((0,T),\mathbb{F})$ satisfying (9) in Itô sense. Furthermore, $Y_{t}$ has
continuous paths.

As argued in [5], if the final condition $X\in L^{\infty}(\mathcal{F}_{T})$
and $g(\cdot,0,0)\equiv 0$ then we have the further regularity
$(Y_{t},Z_{t})\in{L^{\infty}((0,T),\mathbb{F})}\times\mbox{BMO}(\mathbb{P})$.
We need some results about further regularity of the solution when
$g(\cdot,0,0)\not\equiv 0$. Furthermore, the quadratic case has been studied
in [27] and, later, in [48]. In particular, [48] has shown that if
$g(\cdot,0,0)$ is deterministic and $\int_{t}^{T}|g(s,0,0)|ds<+\infty$ then
the solution $(Y_{t},Z_{t})\in
L^{\infty}((0,T);\mathbb{F})\times\mbox{BMO}(\mathbb{P})$. Under Lipschitz
assumptions, we can prove the same thesis weakening the hypotheses on
$g(\cdot,0,0)$. This result will be useful in the following, for what concerns
the theory of BSVIEs.

######  Proposition 4.

Let $X\in L^{\infty}(\mathcal{F}_{T})$ and suppose that
$g:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$
satisfies assumptions a) and b) of Theorem 3 plus c’)
$g(\cdot,0,0)\in\mbox{BMO}(\mathbb{P})$ (called ’standard assumptions’
henceforth).

Then there exists a unique pair of stochastic processes
$(Y_{t},Z_{t})_{t\in[0,T]}\in
L^{\infty}((0,T),\mathbb{F})\times\mbox{BMO}(\mathbb{P})$ satisfying (9) in
Itô sense. Furthermore, $Y_{t}$ has continuous paths.

###### Proof.

By Theorem 3 there exists a unique solution $(Y_{t},Z_{t})_{t\in[0,T]}\in
L^{2}((0,T),\mathbb{F})\times L^{2}((0,T),\mathbb{F})$, and $Y_{t}$ has
continuous paths. We only need to prove the further regularity. By Proposition
3 of [21] we obtain:

$\mathbb{E}\left[\sup_{m\in[t,T]}|y_{m}|^{2}\Big{|}\mathcal{F}_{t}\right]+\mathbb{E}\left[\int_{t}^{T}|Z_{s}|^{2}ds\Big{|}\mathcal{F}_{t}\right]\leq
C\left(\mathbb{E}\left[|X|^{2}|\mathcal{F}_{t}\right]+\mathbb{E}\left[\left.\int_{t}^{T}|g(s,0,0)|^{2}ds\right|\mathcal{F}_{t}\right]\right).$
(10)

As far as $Z_{t}$ regularity is concerned, by equation (10) and by taking the
sup over $t\in[0,T]$ and $\omega\in\Omega$ we get:

$\left\|\sup_{t\in[0,T]}\mathbb{E}\left[\left.\int_{t}^{T}|Z_{s}|^{2}ds\right|\mathcal{F}_{t}\right]\right\|_{\infty}\leq
C\left(\|X\|_{\infty}^{2}+\left\|\sup_{t\in[0,T]}\mathbb{E}\left[\left.\int_{t}^{T}|g(s,0,0)|^{2}ds\right|\mathcal{F}_{t}\right]\right\|_{\infty}\right)\leq
C<+\infty,$ (11)

where here and for the rest of the proof $C>0$ is a constant that varies from
line to line and where the last inequality follows from
$g(\cdot,0,0)\in\mbox{BMO}(\mathbb{P})$. Hence, $Z\in\mbox{BMO}(\mathbb{P})$.
For what concerns $Y_{t}$ we have:

$\mathbb{E}\left[Y_{t}\big{|}\mathcal{F}_{t}\right]=Y_{t}=\mathbb{E}\left[X\big{|}\mathcal{F}_{t}\right]+\mathbb{E}\left[\left.\int_{t}^{T}g(s,Y_{s},Z_{s})ds\right|\mathcal{F}_{t}\right].$

By Jensen’s inequality, Lipschitz assumption and inequalities above, it then
follows that

$\displaystyle|Y_{t}|^{2}$ $\displaystyle\leq
C\left(\|X\|_{\infty}^{2}+\mathbb{E}\left[\int_{t}^{T}\left(|g(s,0,0)|^{2}+|Z_{s}|^{2}\right)ds+\sup_{m\in[t,T]}|y_{m}|^{2}\Big{|}\mathcal{F}_{t}\right]\right)$
$\displaystyle\leq
C\left(\|X\|_{\infty}^{2}+\mathbb{E}\left[\int_{t}^{T}|g(s,0,0)|^{2}\Big{|}\mathcal{F}_{t}\right]\right).$

Taking the supremum, we then obtain:

$\Big{\|}\sup_{t\in[0,T]}|Y_{t}|\Big{\|}_{\infty}^{2}\leq
C\left(\|X\|_{\infty}+\left\|\sup_{t\in[0,T]}\mathbb{E}\left[\int_{t}^{T}|g(s,0,0)|^{2}ds\Big{|}\mathcal{F}_{t}\right]\right\|_{\infty}\right)\leq
C<+\infty.$ (12)

Thus, $Y_{t}\in L^{\infty}((0,T),\mathbb{F})$ and the proof is concluded. ∎

The relation between BSDEs and dynamic risk measures has been deeply studied
in the literature. We address an interested reader to [38, 36, 4, 20, 14],
among others. In particular, concerning cash-subadditive risk measures, El
Karoui and Ravanelli [20] proved that if the driver $g$ satisfies the standard
assumptions (or a quadratic growth assumption), is convex in $(y,z)$ and
decreasing in $y$, then $\rho_{t}(X)\triangleq Y_{t}^{-X}$, with $Y_{t}^{-X}$
being the first component of the solution of the BSDE

$Y_{t}^{-X}=-X+\int_{t}^{T}g(s,Y_{s},Z_{s})ds-\int_{t}^{T}Z_{s}dB_{s},\quad
X\in L^{\infty}(\mathcal{F}_{T}),$ (13)

is a dynamic convex, cash-subadditive, and time-consistent risk measure.
Furthermore, $\rho_{t}$ admits the following dual representation:

$\rho_{t}(X)=\operatorname*{ess\,sup}_{(\beta_{t},\mu_{t})\in\mathcal{A}}\mathbb{E}_{\mathbb{Q}_{t}^{\mu}}\left[\left.e^{-\int_{t}^{T}\beta_{s}ds}(-X)-\int_{t}^{T}e^{-\int_{t}^{s}\beta_{u}du}\,G(s,\beta_{s},\mu_{s})ds\right|\mathcal{F}_{t}\right],$
(14)

where

$\mathbb{E}\left[\frac{d\mathbb{Q}^{\mu}_{t}}{d\mathbb{P}}\Bigg{|}\mathcal{F}_{t}\right]=\mathcal{E}(\mu\cdot
B)_{t},$

$G$ is the Fenchel transformation (or convex conjugate) of $g$, i.e.

$G(t,\beta,\mu)\triangleq\displaystyle\sup_{(y,z)\in\mathbb{R}\times\mathbb{R}^{n}}\left\\{-\beta
y-\langle\mu,z\rangle-g(t,y,z)\right\\},$

and

$\mathcal{A}\triangleq\left\\{(\beta_{t},\mu_{t})_{t\in[0,T]}\
\mathcal{F}_{t}\mbox{-adapted}:G(t,\beta_{t},\mu_{t})<+\infty,\beta_{t}\geq
0,0\leq\beta_{t}+|\mu_{t}|\leq C\mbox{ for any }t\in[0,T]\right\\}.$

By combining Theorems 7.5 in [20] and Proposition 3.6 in [14], it can be
easily checked that if the driver $g$ satisfies the standard assumptions and
convexity, then $G(t,\beta_{t},\mu_{t})=+\infty$ if $\beta_{t}<0$ or
$\beta_{t}+|\mu_{t}|\not\in[0,C]$. Moreover, the $\operatorname*{ess\,sup}$ in
(14) is attained for some
$(\bar{\beta}_{t},\bar{\mu}_{t})_{t\in[0,T]}\in\mathcal{A}$. Hence

$\rho_{t}(X)=\mathbb{E}_{\mathbb{Q}_{t}^{\bar{\mu}}}\left[\left.e^{-\int_{t}^{T}\bar{\beta}_{s}ds}(-X)-\int_{t}^{T}e^{-\int_{t}^{s}\bar{\beta}_{u}du}\,G(s,\bar{\beta}_{s},\bar{\mu}_{s})ds\right|\mathcal{F}_{t}\right].$
(15)

In particular, the following result deals with the sublinear and cash-
subadditive case. We recall that with sublinear it is meant the case where
both subadditivity and positive homogeneity are fulfilled. In terms of the
driver $g$ this corresponds to

\- positive homogeneity in $(y,z)$:

$d\mathbb{P}\times dt\mbox{-a.s., }\ g(t,\alpha y,\alpha z)=\alpha
g(t,y,z)\mbox{ for any }\alpha\geq 0,(y,z)\in\mathbb{R}\times\mathbb{R}^{n};$

\- subadditivity in $(y,z)$:

$d\mathbb{P}\times dt\mbox{-a.s., }\ g(t,y_{1}+y_{2},z_{1}+z_{2})\leq
g(t,y_{1},z_{1})+g(t,y_{2},z_{2})\mbox{ for any
}(y_{1},z_{1}),(y_{2},z_{2})\in\mathbb{R}\times\mathbb{R}^{n}.$

######  Lemma 5.

Suppose that the driver $g$ is decreasing in $y$ and satisfies the standard
assumptions. If, in addition, $g$ is sublinear, then the dynamic risk measure
$\rho_{t}(X)=Y_{t}^{-X}$ induced by (13) is sublinear and admits the dual
representation:

$\rho_{t}(X)=\operatorname*{ess\,sup}_{(\beta_{t},\mu_{t})\in\mathcal{A}}\mathbb{E}_{\mathbb{Q}_{t}^{\mu}}\left[e^{-\int_{t}^{T}\beta_{s}ds}(-X)\Big{|}\mathcal{F}_{t}\right]=\mathbb{E}_{\mathbb{Q}_{t}^{\bar{\mu}}}\left[e^{-\int_{t}^{T}\bar{\beta}_{s}ds}(-X)\Big{|}\mathcal{F}_{t}\right],\mbox{
\ }X\in L^{\infty}(\mathcal{F}_{T}),$ (16)

where $(\bar{\beta},\bar{\mu})\in\mathcal{A}$ is an optimal scenario.

###### Proof.

If $g$ is sublinear then convexity is automatically satisfied, so the dual
representation of $\rho_{t}$ provided in equations (14) and (15) holds for any
$X\in L^{\infty}(\mathcal{F}_{T})$. Moreover, from Theorem 2.3.1(V) of [47] it
follows that the convex conjugate of a positively homogeneous function is $0$
in its domain (i.e. $G(t,\beta,\mu)<+\infty$ implies $G(t,\beta,\mu)=0$).
Finally, sublinearity of $\rho_{t}$ can be easily checked. ∎

####  2.3.2 BSVIEs

somehow related to BSDEs, the family of Backward Stochastic Volterra Integral
Equations (BSVIEs) was studied for the first time in Yong [44]. See also [25,
43] and, for the applications to risk measures, [1, 17, 45]. In the sequel, we
focus on the following (special form of a) BSVIE:

$Y(t)=\varphi(t)+\int_{t}^{T}g(t,s,Y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dB_{s},$
(17)

where $\varphi(t)$ and $g$ are given, while the pair
$(Y(\cdot),Z(\cdot,\cdot))$ is unknown. Note that $\varphi(t)\in
L^{2}(\mathcal{F}_{T})$ is not necessarily $\mathcal{F}_{t}$-measurable but
varies with $t$.

As in the case of standard assumptions for BSDEs, we introduce an analogous
for BSVIEs (see, for instance, [45]). We remind that
$\Delta\triangleq\\{(t,s)\in[0,T]^{2}:t\leq s\\}$. In the sequel,
$g:\Omega\times\Delta\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ will
be required to satisfy the following standard assumptions:

$i)$ $g$ is
$\mathcal{F}_{T}\otimes\mathcal{B}\left(\Delta\times\mathbb{R}\times\mathbb{R}^{n}\right)$-measurable
and $g(t,s,y,z)$ is $(\mathcal{F}_{t})_{t\in[0,T]}$-progressively measurable
on $[t,T]$ for any $(t,y,z)\in[0,T)\times\mathbb{R}\times\mathbb{R}^{n}$;

$ii)$ Uniformly Lipschitz: there exists $L>0$ such that

$|g(t,s,y_{1},z_{1})-g(t,s,y_{2},z_{2})|\leq L(|y_{1}-y_{2}|+|z_{1}-z_{2}|)$

for any
$(t,s)\in\Delta,(y_{1},z_{1}),(y_{2},z_{2})\in\mathbb{R}\times\mathbb{R}^{n}$;

$iii)$
$\mathbb{E}\left[\int_{0}^{T}\left(\int_{t}^{T}|g(t,s,0,0)|ds\right)^{2}dt\right]<+\infty.$

We recall from [46] that a pair
$(Y(\cdot),Z(\cdot,\cdot))\in\mathcal{H}^{1}(0,T)$ is a solution of the BSVIE
(17) if it solves equation (17) for almost all $t\in[0,T]$ in the usual Itô
sense.

######  Theorem 6 (Existence and Uniqueness - see Yong [44]).

Let the driver
$g:\Omega\times\Delta\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$
satisfy the standard assumptions. Then, for any $\varphi\in
L^{2}((0,T),{\mathcal{F}_{T}};\mathbb{R})$, the BSVIE (17) admits a unique
solution in $\mathcal{H}^{2}(0,T)$.

In the following we are mostly interested in BSVIEs of the form:

$Y(t)=\varphi(t)+\int_{t}^{T}g(t,s,Z(t,s))ds-\int_{t}^{T}Z(t,s)dB_{s},\
t\in[0,T],$ (18)

when $g$ does not depend on $y$ and $\varphi\in
L^{\infty}((0,T),\mathcal{F}_{T})$. We show in the next theorem that the
solution $(Y(\cdot),Z(\cdot,\cdot))$ of the BSVIE above has some further
regularity.

######  Theorem 7.

Let $g:\Omega\times\Delta\times\mathbb{R}^{n}\to\mathbb{R}$ be independent of
$y$ and satisfy assumptions i) and ii) above plus the further constraint
$g(\cdot,\cdot,0,0)\in L^{\infty}((0,T);\mbox{BMO}(\mathbb{P}))$, i.e.

$\sup_{t\in[0,T]}\left\|\sup_{r\in[t,T]}\mathbb{E}\left[\int_{r}^{T}|g(t,s,0)|^{2}ds\Big{|}\mathcal{F}_{r}\right]\right\|_{\infty}<+\infty.$

Let $\varphi\in L^{\infty}((0,T),\mathcal{F}_{T})$. Then there exists a unique
solution to BSVIE (18) such that

$(Y(\cdot),Z(\cdot,\cdot))\in L^{\infty}((0,T),\mathbb{F})\times
L^{\infty}((0,T);\mbox{BMO}(\mathbb{P})).$

###### Proof.

Existence and uniqueness of the solution in $\mathcal{H}^{2}(0,T)$ are due to
Theorem 6. Thus, we only need to prove the further regularity of this
solution. To this aim, we consider the following family of BSDEs parameterized
by $t\in[0,T]$:

$\eta(r;t)=\varphi(t)+\int_{r}^{T}g(t,s,\zeta(s;t))ds-\int_{r}^{T}\zeta(s;t)dB_{s},\
\ r\in[t,T].$ (19)

By Proposition 4, for all fixed $t\in[0,T]$ the previous BSDE admits a unique
adapted solution $(\eta(t,\cdot),\zeta(t,\cdot))\in
L^{\infty}((t,T),{\mathbb{F}})\times\mbox{BMO}(\mathbb{P})$. Furthermore, by
(11) we have the estimate:

$\left\|\sup_{r\in[t,T]}\mathbb{E}\left[\int_{r}^{T}|\zeta(s,t)|^{2}ds\Big{|}\mathcal{F}_{r}\right]\right\|_{\infty}\leq
C_{1}\left(\|\varphi(t)\|_{\infty}^{2}+\left\|\sup_{r\in[t,T]}\mathbb{E}\left[\int_{r}^{T}|g(t,s,0)|^{2}ds\Big{|}\mathcal{F}_{r}\right]\right\|_{\infty}\right)\leq
C_{2}<+\infty$

for some constants $C_{1},C_{2}>0$ that do not depend on $(t,s)\in\Delta$. It
then follows that

$\displaystyle\sup_{t\in[0,T]}\left\|\sup_{r\in[t,T]}\mathbb{E}\left[\left.\int_{r}^{T}|\zeta(s,t)|^{2}ds\right|\mathcal{F}_{r}\right]\right\|_{\infty}$
$\displaystyle\leq
C_{3}\left(\sup_{t\in[0,T]}\|\varphi(t)\|_{\infty}^{2}+\sup_{t\in[0,T]}\left\|\sup_{r\in[t,T]}\mathbb{E}\left[\left.\int_{r}^{T}|g(t,s,0)|^{2}ds\right|\mathcal{F}_{r}\right]\right\|_{\infty}\right)\leq
C_{4}<+\infty,$

for some constants $C_{3},C_{4}>0$, where the last inequality follows from
$\varphi\in L^{\infty}((0,T),\mathcal{F}_{T})$ and from $g(\cdot,\cdot,0,0)\in
L^{\infty}((0,T);\mbox{BMO}(\mathbb{P}))$. For any $t\in[0,T]$, (12) also
gives the estimate:

$\Big{\|}\sup_{r\in[t,T]}|\eta(r;t)|\Big{\|}_{\infty}^{2}\leq
C_{5}\left(\|\varphi(t)\|_{\infty}+\left\|\sup_{r\in[t,T]}\mathbb{E}\left[\int_{r}^{T}|g(t,s,0)|^{2}ds\Big{|}\mathcal{F}_{t}\right]\right\|_{\infty}\right)\leq
C_{6}<+\infty$

for some $C_{5},C_{6}>0$. Taking the supremum over $t\in[0,T]$, we can
conclude $\eta\in L^{\infty}((0,T);L^{\infty}((t,T),\mathbb{F}))$. The thesis
follows by considering

$Y(t)\triangleq\eta(t;t),\ Z(t,s)\triangleq\zeta(s;t),\quad\mbox{ for
}(t,s)\in\Delta.$

∎

We are now ready to present a version of the Comparison Theorem for BSVIEs.
This result will be used in the ensuing parts of the paper. More details about
Comparison Theorems for BSVIEs can be found in [43]. Differently from [43], we
require $(y,z)\in\mathbb{R}\times\mathbb{R}^{n}$ instead of
$(y,z)\in\mathbb{R}^{n}\times\mathbb{R}^{n\times m}$, which allows us to
weaken the hypotheses regarding the differentiability of $g$.

######  Proposition 8 (Comparison Theorem).

For $i=1,2$, let $g_{i}:\Omega\times\Delta\times\mathbb{R}^{n}\to\mathbb{R}$
be independent of $y$, satisfy the assumptions of Theorem 7 and

$g_{1}(t,s,z)\leq g_{2}(t,s,z)\mbox{ a.s., a.e.
}(t,s,z)\in\Delta\times\mathbb{R}^{n}.$

Assume also that $\varphi_{i}\in L^{\infty}((0,T),\mathcal{F}_{T})$ have
continuous paths and satisfy $\varphi_{1}(t)\leq\varphi_{2}(t)$, a.s., a.e.
$t\in[0,T]$. Then the solutions $(Y_{i}(\cdot),Z_{i}(\cdot,\cdot))$ of the
BSVIEs (18) with parameters $(g_{i},\varphi_{i})$ satisfy:

$Y_{1}(t)\leq Y_{2}(t),\mbox{ a.s., a.e. }t\in[0,T].$

###### Proof.

This proof is similar to that of Proposition 3.3 in [43].

Fixing $t\in[0,T]$, let us consider again the families of BSDEs parameterized
by $t$:

$\eta^{i}(r;t)=\varphi^{i}(t)+\int_{r}^{T}g(t,s,\zeta^{i}(s;t))ds-\int_{r}^{T}\zeta^{i}(s;t)dB_{s},\
\ r\in[t,T],$

for $i=1,2$. Since $g^{i}(t,s,z)$ and $\varphi^{i}(t)$ satisfy all the
hypotheses of Theorem 2.2 of [19],

$\eta^{1}(r;t)\leq\eta^{2}(r,t),\quad\mbox{ for any }r\in[t,T],\mbox{ a.s.}$
(20)

Set now

$Y^{i}(t)=\eta^{i}(t;t)\mbox{ and }Z^{i}(r;t)=\zeta^{i}(r;t),\quad\mbox{ for
any }(t,r)\in\Delta,$

which solve the corresponding BSVIE (18). Letting $r\to t^{+}$ in equation
(20) and using the pathwise continuity of $\eta(\cdot;t)$ (see Theorem 3), we
obtain $Y^{1}(t)\leq Y^{2}(t)$ a.s., $t\in[0,T]$. ∎

##  3 Capital allocation rules for cash-subadditive risk measures

This section contains our main results on capital allocation rules induced by
cash-subadditive risk measures where the approach of [33] is extended to the
non cash-additive case. First of all, we prove the one-to-one correspondence
between cash-subadditive risk measures and capital allocation rules satisfying
suitable properties. Afterwards, we consider the setting where risk measures
are induced by BSDEs.

Unless otherwise stated, in the sequel we will focus on dynamic risk measures
(and capital allocations) defined on $L^{\infty}(\mathcal{F}_{T})$.

###  3.1 Axiomatic approach

Let $\rho_{t}$ be a dynamic risk measure as in (2) where we assume that for
each $X\in L^{\infty}(\mathcal{F}_{T})$ the $\operatorname*{ess\,sup}$ is
attained for some
$({D}^{X}_{t},\mathbb{Q}^{X}_{t})\in\mathcal{D}\times\mathcal{Q}$ (not
necessarily unique). Define now

$\Lambda^{sub}_{t}(X,Y)\triangleq\mathbb{E}_{\mathbb{Q}_{t}^{Y}}\left[-D^{Y}_{t}X\Big{|}\mathcal{F}_{t}\right]-c_{t}(D^{Y}_{t}\mathbb{Q}^{Y}_{t}),\quad
X,Y\in L^{\infty}(\mathcal{F}_{T}),$ (21)

where

$(D^{Y}_{t},\mathbb{Q}^{Y}_{t})\in\operatorname*{arg\,ess\,max}_{(D_{t},\mathbb{Q}_{t})\in\mathcal{D}\times\mathcal{Q}}\left\\{\mathbb{E}_{\mathbb{Q}}\left[-D_{t}Y\Big{|}\mathcal{F}_{t}\right]-c_{t}(D_{t}\mathbb{Q}_{t})\right\\}.$

It is easy to check that the family $(\Lambda_{t}^{sub})_{t\in[0,T]}$ is a
dynamic CAR that henceforth will be called dynamic cash-subadditive
subdifferential CAR associated to $\rho_{t}$. To be more precise,
$(\Lambda_{t}^{sub})_{t\in[0,T]}$ is a family of CARs defined via optimal
scenarios. So, when we refer to $(\Lambda_{t}^{sub})_{t\in[0,T]}$ we mean one
of the possible CARs corresponding to a specific scenario
$(D_{t},\mathbb{Q}_{t})\in\mathcal{D}\times\mathcal{Q}$.

Note that the reason why $\Lambda_{t}^{sub}$ is called a subdifferential CAR
is because, under suitable hypotheses (see Proposition 9 below), a scenario
$(D_{t},\mathbb{Q}_{t})$ is optimal in the representation of $\rho_{t}(X)$ if
and only if $D_{t}\cdot\mathbb{Q}_{t}$ belongs to the subdifferential of
$\rho_{t}$ at $X$, defined as

$\partial\rho_{t}(X)\triangleq\\{Z\in L^{1}_{+}(\mathcal{F}_{T}):\
\rho_{t}(Y)\geq\rho_{t}(X)+\mathbb{E}[-Z(Y-X)|\mathcal{F}_{t}]\mbox{ for every
}Y\in L^{\infty}(\mathcal{F}_{T})\\}.$ (22)

As a consequence, more than one optimal scenario is possible when
$\rho_{t}(X)$ is only subdifferentiable, while if $\rho_{t}(X)$ is also
Gateaux differentiable then the subdifferential is a singleton and there
exists a unique $D_{t}\cdot\mathbb{Q}_{t}$ with $(D_{t},\mathbb{Q}_{t})$ being
an optimal scenario. We recall (see, e.g., [47]) that the directional
derivative of $\rho_{t}$ at $X$ in the direction $Y$ is defined (if there
exists) as

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

while $\rho_{t}(X)$ is said to be Gateaux differentiable at $X$ if

$D_{\rho}^{+}(Y,X)\triangleq\lim_{h\downarrow
0}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h}$

exists for any $Y\in L^{\infty}(\mathcal{F}_{T})$ and there exists
$\nabla\rho_{t}(X)\in L^{1}_{+}(\mathcal{F}_{T})$ such that
$D_{\rho}^{+}(Y,X)=\mathbb{E}_{\mathbb{P}}[-Y\nabla\rho_{t}(X)]$ for any $Y\in
L^{\infty}(\mathcal{F}_{T})$. In such a case, $\nabla\rho_{t}(X)$ is unique
and is called the Gateaux derivative. Furthermore, also
$\nabla\rho_{t}(Y,X)\triangleq\lim_{h\to
0}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h}$ exists and
$\partial\rho_{t}(X)=\\{\nabla\rho_{t}(X)\\}$. Note that the definition of
Gateaux differentiability is not universal.

######  Proposition 9.

Let $\rho_{t}:L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$ be a
dynamic cash-subadditive and continuous from above risk measure and suppose
that the $\operatorname*{ess\,sup}$ in (2) is attained for a fixed $X\in
L^{\infty}(\mathcal{F}_{T})$.

A scenario $(D^{X}_{t},\mathbb{Q}^{X}_{t})\in\mathcal{D}\times\mathcal{Q}$ is
optimal if and only if the density $Z_{t}^{X}\triangleq
D_{t}^{X}\mathbb{Q}_{t}^{X}$ is an element of $\partial\rho_{t}(X)$. Moreover,
if there exists $D_{\rho}(Y,X)$ at every $Y\in L^{\infty}(\mathcal{F}_{T})$
(in particular, $\rho_{t}$ is Gateaux differentiable at $X$) then
$\partial\rho_{t}(X)$ is a singleton.

###### Proof.

Let $X\in L^{\infty}(\mathcal{F}_{T})$ be fixed and assume that
$(D_{t}^{X},\mathbb{Q}_{t}^{X})$ is an optimal scenario in (2). Then the
associated $\lambda_{t}^{X}\triangleq D^{X}_{t}\mathbb{Q}^{X}_{t}$ is an
absolutely continuous sub-probability with respect to $\mathbb{P}$. It then
follows that, for any $Y\in L^{\infty}(\mathcal{F}_{T})$,

$\displaystyle\rho_{t}(Y)-\rho_{t}(X)$ $\displaystyle=$
$\displaystyle\operatorname*{ess\,sup}_{\lambda_{t}\in\widetilde{\mathcal{S}}}\\{\mathbb{E}_{\lambda_{t}}[-Y\big{|}\mathcal{F}_{t}]-c_{t}(\lambda_{t})\\}-\left(\mathbb{E}_{\lambda_{t}^{X}}[-X\big{|}\mathcal{F}_{t}]-c_{t}(\lambda_{t}^{X})\right)$
(23) $\displaystyle\geq$
$\displaystyle\mathbb{E}_{\lambda_{t}^{X}}[-(Y-X)\big{|}\mathcal{F}_{t}],$

where $\widetilde{\mathcal{S}}\triangleq\left\\{\lambda_{t}\mbox{ sub-
probability}:\lambda_{t}=D_{t}\mathbb{Q}_{t}\mbox{ for some
}(D_{t},\mathbb{Q}_{t})\in\mathcal{D}\times\mathcal{Q}\right\\}$. Hence, the
density of the sub-probability $\lambda_{t}^{X}$ defined by
$Z^{X}_{t}\triangleq\frac{d\lambda_{t}^{X}}{d\mathbb{P}}=D_{t}^{X}\frac{d\mathbb{Q}^{X}_{t}}{d\mathbb{P}}$
is an element of the subdifferential $\partial\rho_{t}(X)$.

Conversely, we are now going to prove that any element of the subdifferential
can be identified with an optimal scenario $(D^{X}_{t},\mathbb{Q}^{X}_{t})$.
We first prove that, given $X\in L^{\infty}(\mathcal{F}_{T})$, if
$Z\in\partial\rho_{t}(X)$ then its $L^{1}$-norm is less or equal to $1$.
Indeed, by taking $Y=X-m$ with $m\in\mathbb{R}_{+}$ and using cash-
subadditivity, it follows that

$\rho_{t}(X)+m\geq\rho_{t}(Y)\geq\rho_{t}(X)+m\,\mathbb{E}[Z\big{|}\mathcal{F}_{t}],$

where the last inequality is due to $Z\in\partial\rho_{t}(X)$. This implies
that $0\leq\mathbb{E}[Z\big{|}\mathcal{F}_{t}]\leq 1$, hence that
$0\leq\mathbb{E}[Z]\leq 1$ and $Z$ is a density corresponding to a sub-
probability $\lambda$ through $Z\triangleq\frac{d\lambda}{d\mathbb{P}}$. From
now on, we identify the sub-probability $\lambda$ with its density, so any
element of $\partial\rho_{t}(X)$ is understood as a sub-probability via this
identification. By Proposition 6 in [31], we know that every sub-probability
$\lambda_{t}$ absolutely continuous w.r.t. $\mathbb{P}$ such that
$c_{0,t}(\lambda_{t})<+\infty$ can be decomposed as
$\lambda_{t}=D_{t}\mathbb{Q}_{t}$ with
$(D_{t},\mathbb{Q}_{t})\in\mathcal{D}\times\mathcal{Q}$, where $c_{0,t}$ is
the minimal penalty function of the static cash-subadditive (continuous from
above) risk measure $\rho_{0,t}(X)\triangleq\mathbb{E}[\rho_{t}(X)]$ (see
Proposition 6 in [31]). If $\lambda_{t}^{X}\in\partial\rho_{t}(X)$ (meaning
that $Z_{t}^{X}\in\partial\rho_{t}(X)$) then, by applying the
$\mathbb{P}$-expectation to the inequality in (23) and by using the fact that
$\mathbb{E}_{\mathbb{P}}[\mathbb{E}_{\lambda^{X}_{t}}[\cdot\big{|}\mathcal{F}_{t}]]=\mathbb{E}_{\lambda_{t}^{X}}[\cdot]$,

$\mathbb{E}_{\lambda_{t}^{X}}[-Y]-\rho_{0,t}(Y)\leq\mathbb{E}_{\lambda_{t}^{X}}[-X]-\rho_{0,t}(X)\
\mbox{for every }Y\in L^{\infty}(\mathcal{F}_{t}).$ (24)

By the representation for the penalty term of a static cash-subadditive risk
measure (see [20]) and by (24), we have:

$c_{0,t}(\lambda_{t}^{X})=\sup_{Y\in
L^{\infty}(\mathcal{F}_{T})}\left\\{\mathbb{E}_{\lambda_{t}^{X}}[-Y]-\rho_{0,t}(Y)\right\\}=\mathbb{E}_{\lambda_{t}^{X}}[-X]-\rho_{0,t}(X)<+\infty.$

So far, we have proved that $\lambda_{t}^{X}$ can be decomposed as
$\lambda_{t}^{X}=D_{t}^{X}\mathbb{Q}^{X}_{t}$. Now, by repeating the same
arguments as above by replacing $\rho_{0,t}$ with $\rho_{t}$, it holds that

$c_{t}(D_{t}^{X}\mathbb{Q}^{X}_{t})=\mathbb{E}_{\mathbb{Q}^{X}_{t}}[-D_{t}^{X}X\big{|}\mathcal{F}_{t}]-\rho_{t}(X)$

and, consequently, that
$(D^{X}_{t},\mathbb{Q}^{X}_{t})\in\mathcal{D}\times\mathcal{Q}$ is an optimal
scenario.

Finally, it remains to prove that, under the existence of the directional
derivative at every direction, $\partial\rho_{t}(X)$ is a singleton. Assume
that there exists $D_{\rho}(Y,X)$ at every $Y\in L^{\infty}(\mathcal{F}_{T})$.
By proceeding similarly to Remark 3 in [33], we have

$\lim_{h\downarrow 0}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h}\geq\lim_{h\downarrow
0}\frac{\mathbb{E}_{\lambda_{t}^{X}}[-hY\big{|}\mathcal{F}_{t}]}{h}=\mathbb{E}_{\lambda_{t}^{X}}[-Y\big{|}\mathcal{F}_{t}],$
(25)

and, analogously, for the left-limit:

$\lim_{h\uparrow 0}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h}\leq\lim_{h\uparrow
0}\frac{\mathbb{E}_{\lambda_{t}^{X}}[-hY\big{|}\mathcal{F}_{t}]}{h}=\mathbb{E}_{\lambda_{t}^{X}}[-Y\big{|}\mathcal{F}_{t}],$
(26)

where inequalities follow from the dual representation of the risk measure
$\rho_{t}$ and from the assumption that
$\lambda_{t}^{X}=D^{X}_{t}\mathbb{Q}^{X}_{t}$, with
$(D^{X}_{t},\mathbb{Q}^{X}_{t})\in\mathcal{D}\times\mathcal{Q}$, is an optimal
scenario. Existence of the directional derivative at every direction, (25) and
(26) then imply that

$\nabla\rho_{t}(X;Y)=\lim_{h\to
0}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h}=\mathbb{E}_{\lambda_{t}^{X}}[-Y\big{|}\mathcal{F}_{t}].$

Suppose now that there exist two optimal scenarios
$\lambda_{t,1}^{X},\lambda_{t,2}^{X}\in\partial\rho_{t}(X)$. In this case, it
would then hold that

$\nabla\rho_{t}(X;Y)=\mathbb{E}_{\lambda_{t,1}^{X}}[-Y\big{|}\mathcal{F}_{t}]=\mathbb{E}_{\lambda_{2,t}^{X}}[-Y\big{|}\mathcal{F}_{t}]\quad\mbox{
for any }Y\in L^{\infty}(\mathcal{F}_{T}),$

which implies that the two densities coincide almost surely, hence that
$\partial\rho_{t}(X)$ is a singleton. ∎

Similarly as proved in [33] for the cash-additive case, the following
correspondence between risk measures and CARs holds also in the cash-
subadditive case.

######  Proposition 10.

a) If $\Lambda_{t}:L^{\infty}(\mathcal{F}_{T})\times
L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$, with $t\in[0,T]$,
is a dynamic CAR satisfying no-undercut, monotonicity, weak-convexity,
normalization, 1-time-consistent (resp. time-consistent), and cash-
subadditivity, then $\rho_{t}(X)\triangleq\Lambda_{t}(X,X)$ is a weak time-
consistent (resp. time-consistent) convex cash-subadditive risk measure.

b) Conversely, if $\rho_{t}$ is a dynamic convex cash-subadditive risk
measures that can be represented as in (2) with the $\operatorname*{ess\,sup}$
attained, then there exists at least a no-undercut, monotone, weakly-convex
and 1-cash-subadditive dynamic CAR $(\Lambda_{t})_{t\in[0,T]}$ which also
satisfies sub-allocation. In particular, $\Lambda_{t}^{sub}$ is a possible CAR
of this kind.

###### Proof.

a) The proof of normalization, convexity, monotonicity, and weak time-
consistency (resp. time-consistency) follows immediately from Proposition 5 in
[33]. The cash-subadditivity of $\rho_{t}$ is easily given by

$\rho_{t}(X+m_{t})=\Lambda_{t}(X+m_{t},X+m_{t})\geq\Lambda_{t}(X,X)-m_{t}=\rho_{t}(X)-m_{t}$

true for any $X\in L^{\infty}(\mathcal{F}_{T}),m_{t}\in
L^{\infty}_{+}(\mathcal{F}_{t})$, where the inequality is due to cash-
subadditivity of $\Lambda_{t}$.

b) We are now going to prove the existence of such a $\Lambda_{t}$ by checking
that, e.g., the subdifferential CAR $\Lambda_{t}^{sub}$ defined in (21)
satisfies all the properties required. First of all,
$\Lambda_{t}^{sub}(X,X)=\rho_{t}(X)$ for any $X\in
L^{\infty}(\mathcal{F}_{T}),t\in[0,T]$, by definition of $\Lambda_{t}^{sub}$.
Monotonicity, weakly-convexity, sub-allocation, and no-undercut follow
similarly as in Proposition 3 of [33]. Concerning 1-cash-additivity, it holds
that, for any $m_{t}\in L^{\infty}_{+}(\mathcal{F}_{t}),X,Y\in
L^{\infty}(\mathcal{F}_{T})$,

$\displaystyle\Lambda_{t}^{sub}(X+m_{t},Y)$ $\displaystyle=$
$\displaystyle\mathbb{E}_{\mathbb{Q}_{t}^{Y}}\left[-D^{Y}_{t}(X+m_{t})\Big{|}\mathcal{F}_{t}\right]-c_{t}(D^{Y}_{t}\mathbb{Q}^{Y}_{t})$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\mathbb{Q}_{t}^{Y}}\left[-D^{Y}_{t}X\Big{|}\mathcal{F}_{t}\right]-m_{t}\,\mathbb{E}_{\mathbb{Q}_{t}^{Y}}[D_{t}Y|\mathcal{F}_{t}]-c_{t}(D^{Y}_{t}\mathbb{Q}^{Y}_{t})$
$\displaystyle\geq$
$\displaystyle\mathbb{E}_{\mathbb{Q}_{t}^{Y}}\left[-D^{Y}_{t}X\Big{|}\mathcal{F}_{t}\right]-m_{t}-c_{t}(D^{Y}_{t}\mathbb{Q}^{Y}_{t})$
$\displaystyle=$ $\displaystyle\Lambda_{t}^{sub}(X,Y)-m_{t},$

where the first equality is due to the $\mathcal{F}_{t}$-measurability of
$m_{t}$, while the inequality follows from $0\leq D_{t}\leq 1$. The proof is
therefore complete. ∎

Note that while time-consistency of $\Lambda_{t}$ implies that of $\rho_{t}$,
the converse implication does not necessarily hold true. In the particular
case of cash-additive risk measures induced by BSDEs, however, the
corresponding CARs are time-consistency, as shown in [33].

###  3.2 Capital allocation rules and backward equations

We focus now our attention on cash-subadditive risk measures induced by BSDEs.
Following El Karoui and Ravanelli [20], then, we consider a BSDE of the form:

$\rho_{t}(X)=-X+\int_{t}^{T}g(s,\rho_{s}(X),Z^{\rho}_{s})ds-\int_{t}^{T}Z^{\rho}_{s}\,dB_{s},\quad
X\in L^{\infty}(\mathcal{F}_{T}),$ (27)

where the driver $g(s,y,z)$ is convex in $(y,z)$, decreasing in $y$ and
satisfies the standard assumptions. Throughout this section we require that,
in the dual representation of $\rho_{t}$, the optimal scenarios
$(D_{t},\mathbb{Q}_{t})$ belong to $\mathcal{D}^{\prime}\times\mathcal{Q}$,
without the further restriction $D_{t}\in\mathcal{D}$.

Inspired by what done in [33] for the cash-additive case (where $g$ does not
depend on $y$), we investigate now the case of dynamic subdifferential CARs
induced by these cash-subadditive risk measures. Differently from the cash-
additive case where BSDEs (or BSVIEs) were preserved when moving from risk
measures to capital allocations, we show that this happens no more in the
cash-subadditive case. For simplicity, we always work with one of the possible
optimal scenarios, in order to restrict the family of
$(\Lambda_{t}^{sub})_{t\in[0,T]}$ to one of its element.

We start with the following motivating example.

######  Example 11.

Following the Example 7.2 in [20], we consider the driver

$g_{\beta}(t,y,z)=\sup_{r_{t}\leq\beta_{t}\leq
R_{t}}(-\beta_{t}y)=-\bar{\beta}_{t}y,$

where $(\beta_{t})_{t\in[0,T]}$ is a given ambiguous discount rate with $0\leq
r_{t}\leq\beta_{t}\leq R_{t}$ with $(r_{t})_{t\in[0,T]}$ and
$(R_{t})_{t\in[0,T]}$ being two adapted and bounded processes. It is well
known that the BSDE (27) with driver $g_{\beta}$ admits as a unique solution:

$\rho_{t}(Y)=\mathbb{E}_{\mathbb{P}}\left[\left.e^{-\int_{t}^{T}\bar{\beta}_{s}ds}(-Y)\right|\mathcal{F}_{t}\right].$
(28)

It then follows that $(\mathbb{P},D^{Y}_{t}\triangleq
e^{-\int_{t}^{T}\bar{\beta}_{s}ds})$ is an optimal scenario for $\rho_{t}(Y)$
with $c_{t}(D_{t}\mathbb{P})=0$ and, by definition,

$\Lambda^{sub}_{t}(X,Y)=\mathbb{E}_{\mathbb{P}}[\left.-D_{t}^{Y}X\right|\mathcal{F}_{t}].$

Note that here $D^{Y}_{t}$ is not necessarily adapted, but only satisfies
$D^{Y}_{t}\in[0,1]$.

By using the Martingale Representation Theorem, then, for any fixed
$t\in[0,T]$ there exists a unique predictable process
$Z^{\Lambda^{sub}}(t,\cdot)=Z^{\Lambda^{s}}(t,\cdot)$ defined on $[t,T]$ such
that:

$\Lambda^{sub}_{t}(X,Y)=-D^{Y}_{t}X-\int_{t}^{T}Z^{\Lambda^{s}}(t,s)ds.$ (29)

Surprisingly, starting from a cash-subadditive risk measure $\rho_{t}(Y)$
induced by a BSDE, the dynamics of $\Lambda_{t}^{sub}(X,Y)$ can be only
written in terms of a BSVIE which, by Theorem 6, admits a unique solution.
This can be explained by the fact that, because of the discounting factor
$D^{Y}_{t}$, also the final condition in (29) depends on $t$.

The previous example underlines how, when dropping cash-additivity, it may be
possible to obtain a CAR that does not follow a classical BSDE, even when the
associated risk measure is induced by a BSDE. This is different from the cash-
additive case and explains why we investigate which kind of backward equation
is followed by a subdifferential CAR associated to a cash-subadditive risk
measure induced by a BSDE.

The impossibility of obtaining $\Lambda_{t}(X,Y)$ through a BSDE suggests a
lack of time-consistency for CARs induced via cash-subadditive risk measures:
this is not necessarily a drawback. Indeed, as argued in Yong [44, 45] and the
references therein, time-inconsistent preferences exist in real markets, so
the time-inconsistency of CARs driven by BSVIEs can be used to model investors
with such preferences. Moreover, the impossibility to reduce the dynamics of
CARs to a BSDE is not completely new: a similar phenomenon, indeed, happens in
the case of standard expected utility (see [1, 44, 45]).

We are now ready to provide the dynamics of the subdifferential CAR generated
by a cash-subadditive risk measure with a generic driver $g$.

######  Proposition 12.

Let $\rho_{t}:L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$ be a
dynamic cash-subadditive convex risk measure induced by a BSDE as in (27)
whose the driver $g$ is convex in $(y,z)$, decreasing in $y$ and satisfies the
standard assumptions. Then, for any $X,Y\in L^{\infty}(\mathcal{F}_{T})$, the
subdifferential CAR defined as in (21) is the first component of the solution
to the following BSVIE that admits a unique solution
$(\Lambda^{sub},Z^{\Lambda^{s}})\in L^{\infty}((0,T),\mathbb{F})\times
L^{\infty}((0,T);\mbox{BMO}(\mathbb{P}))$:

$\Lambda^{sub}_{t}(X,Y)=-D_{t}X+\int_{t}^{T}g_{\Lambda^{s}}(t,s,\rho_{s}(Y),Z_{s}^{\rho(Y)},Z^{\Lambda^{s}}(t,s))ds-\int_{t}^{T}Z^{\Lambda^{s}}(t,s)\,dB_{s},$
(30)

where
$g_{\Lambda^{s}}:\Omega\times\Delta\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$
is defined by:

$g_{\Lambda^{s}}(t,s,\rho_{s}(Y),Z_{s}^{\rho(Y)},Z^{\Lambda^{s}}(t,s))\triangleq
D_{t,s}\left[-g(s,\rho_{s}(Y),Z^{\rho(Y)}_{s})-\beta^{Y}_{s}\rho_{s}(Y)-\mu_{s}^{Y}Z^{\rho}_{s}\right]+\mu^{Y}_{s}Z^{\Lambda^{s}}(t,s),$
(31)

where $D_{t}\triangleq\exp\left\\{-\int_{t}^{T}\beta^{Y}_{s}ds\right\\}$ and
$D_{t,s}\triangleq\exp\left\\{-\int_{t}^{s}\beta^{Y}_{u}du\right\\}$ with
$(\beta^{Y}_{t},\mu^{Y}_{t})$ being (one of) the optimal scenario in the
representation (15) of $\rho_{t}(Y)$, identifying $\mu_{t}^{Y}$ with the
corresponding $\mathbb{Q}_{t}^{\mu^{Y}}$.

Furthermore, $\Lambda_{t}^{sub}(X,Y)$ satisfies no-undercut, monotonicity,
1-cash-subadditivity, weakly-convexity, and sub-allocation. Moreover, the sub-
probability $\lambda_{t}^{Y}\triangleq
e^{-\int_{t}^{T}{\beta}^{Y}_{s}ds}\,\mathbb{Q}^{{\mu}^{Y}}_{t}$ belongs to
$\partial\rho_{t}(Y)$.

###### Proof.

First of all, the BSVIE (30) admits a unique solution in
$L^{\infty}((0,T),\mathbb{F})\times L^{\infty}((0,T);\mbox{BMO}(\mathbb{P}))$
because the driver $g_{\Lambda^{s}}$ satisfies all the assumptions of Theorem
7. Indeed, the $(\mathcal{F}_{t})_{t\in[0,T]}$-progressive measurability
follows from the same property of $g(t,y,z)$ and
$(\beta^{Y}_{t},\mu^{Y}_{t})$, while the uniform Lipschitzianity w.r.t.
$Z^{\Lambda}$ follows from the boundedness of $\mu^{Y}_{t}$. Moreover, we
have:

$|g^{\Lambda^{s}}(t,s,\rho_{s}^{Y},Z^{\rho(Y)}_{s})|^{2}\leq
C(|g(s,0,0)|^{2}+|Z_{s}^{\rho(Y)}|^{2}+1),$

where we used $D_{t,s}\in L^{\infty}((0,T),\mathcal{F}_{s})$, Lipschitzianity
of $g$ and boundedness of $(\beta_{t}^{Y},\mu^{Y}_{t})$ and $\rho_{t}(Y)$. By
integrating and taking the expectation w.r.t. $\mathcal{F}_{r}$ and the
supremum over $r\in[t,T]$,

$\sup_{r\in[t,T]}\mathbb{E}\left[\left.\int_{r}^{T}|g^{\Lambda^{s}}(t,s,\rho_{s}^{Y},Z^{\rho(Y)}_{s})|^{2}ds\right|\mathcal{F}_{r}\right]\leq
C_{1}\sup_{r\in[t,T]}\mathbb{E}\left[\left.\int_{r}^{T}\left(|g(s,0,0)|^{2}+|Z_{s}^{\rho(Y)}|^{2}+1\right)ds\right|\mathcal{F}_{r}\right]\leq
C_{2}$

for some $C_{1},C_{2}>0$, where the last inequality follows from
$g(\cdot,0,0),Z_{\cdot}^{\rho(Y)}\in\mbox{BMO}(\mathbb{P})$ (see Proposition
4). Since $C_{1}$ and $C_{2}$ do not depend on $t$ or $\omega$,

$\sup_{t\in[0,T]}\left\|\sup_{r\in[t,T]}\mathbb{E}\left[\left.\int_{r}^{T}|g^{\Lambda^{s}}(t,s,\rho_{s}^{Y},Z^{\rho(Y)}_{s})|^{2}ds\right|\mathcal{F}_{r}\right]\right\|_{\infty}\leq
C_{2}<+\infty,$

thus $g_{\Lambda^{s}}$ satisfies the assumptions of Theorem 7.

The rest of the proof is similar to that used in the proof of Proposition 6 in
[33]. By the dual representation of $\rho_{t}(Y)$ with optimal scenario
$(\beta^{Y}_{t},\mu^{Y}_{t})$,

$\rho_{t}(Y)=\mathbb{E}_{\mathbb{Q}^{\mu^{Y}}}\left[\left.e^{-\int_{t}^{T}\beta^{Y}_{s}ds}\,(-Y)-\int_{t}^{T}e^{-\int_{t}^{s}\beta^{Y}_{u}du}\,\,G(s,\beta^{Y}_{s},\mu^{Y}_{s})ds\right|\mathcal{F}_{t}\right].$

By definition, then,

$\Lambda^{sub}_{t}(X,Y)=\mathbb{E}_{\mathbb{Q}^{\mu^{Y}}}\left[\left.e^{-\int_{t}^{T}\beta^{Y}_{s}ds}\,(-X)-\int_{t}^{T}e^{-\int_{t}^{s}\beta^{Y}_{u}du}\,\,G(s,\beta^{Y}_{s},\mu^{Y}_{s})ds\right|\mathcal{F}_{t}\right],$
(32)

where the explicit expression of the penalty term appears. By the Martingale
Representation Theorem, for any fixed $t\in[0,T]$ there exists a predictable
process $Z^{\Lambda^{s}}(t,\cdot)$ such that:

$\Lambda^{sub}_{t}(X,Y)=e^{-\int_{t}^{T}\beta^{Y}_{s}ds}(-X)-\int_{t}^{T}e^{-\int_{t}^{s}\beta^{Y}_{u}du}G(s,\beta^{Y}_{s},\mu^{Y}_{s})\,ds-\int_{t}^{T}Z^{\Lambda^{s}}(t,s)\,dB^{\mu^{Y}}_{s},$

where $dB_{t}^{\mu^{Y}}=dB_{t}+\mu^{Y}_{t}dt$. Using Lemma 7.4 (ii) of [20]
and making the definition of $B_{t}^{\mu^{Y}}$ explicit, the first statement
of the result follows.

By construction, $\Lambda_{t}^{sub}$ is a CAR since
$\Lambda^{sub}_{t}(Y,Y)=\rho_{t}(Y)$ for any $Y\in
L^{\infty}(\mathcal{F}_{T})$. No-undercut, monotonicity, 1-cash-subadditivity,
weakly-convexity, and sub-allocation are consequences of Proposition 10.

It remains to prove that $\lambda_{t}^{Y}\in\partial\rho_{t}(Y)$. To this aim,
we remind (see [20, 47]) that an element
$(\beta,\mu)\in\mathbb{R}\times\mathbb{R}^{n}$ attains the sup in the Fenchel
transformation of $g$ (i.e. $G(t,\beta,\mu)=-g(t,y,z)-\beta y-\mu\cdot z$) if
and only if $(\beta,\mu)\in\partial
g(t,y,z)\triangleq\\{(\bar{\beta},\bar{\mu})\in\mathbb{R}\times\mathbb{R}^{n}:g(t,v,w)\geq
g(t,y,z)-\bar{\beta}(v-y)-\bar{\mu}(w-z)\,\mbox{ for every
}(v,w)\in\mathbb{R}\times\mathbb{R}^{n}\\}$. Fix now $X\in
L^{\infty}(\mathcal{F}_{T})$. For every $Y\in L^{\infty}(\mathcal{F}_{T})$ it
then holds that

$\displaystyle-d(\rho_{t}(Y)-\rho_{t}(X))$ $\displaystyle=$
$\displaystyle\left[g(t,\rho_{t}(Y),Z^{Y}_{t})-g(t,\rho_{t}(X),Z^{X}_{t})\right]dt-(Z^{Y}_{t}-Z_{t}^{X})dB_{t}$
$\displaystyle\geq$
$\displaystyle-\beta^{X}_{t}(\rho_{t}(Y)-\rho_{t}(X))dt-\mu^{X}_{t}(Z^{Y}_{t}-Z_{t}^{X})dt-(Z^{Y}_{t}-Z_{t}^{X})dB_{t}$
$\displaystyle=$
$\displaystyle-\beta^{X}_{t}(\rho_{t}(Y)-\rho_{t}(X))dt-(Z^{Y}_{t}-Z_{t}^{X})dB_{t}^{\mu^{X}}$

with final condition $-(Y-X)$, where $dB_{t}^{\mu^{X}}\triangleq
dB_{t}+\mu^{X}_{t}dt$ and where the inequality is due to the definition of
subdifferential of $g$ and to Theorem 7.1 in [20]. Consider now the following
BSDE:

$-d\xi_{t}=-\beta^{X}_{t}\xi_{t}\,dt-\Gamma_{t}\,dB^{\mu^{X}}_{t}\mbox{ and
}\xi_{T}=-(Y-X),$

with solution

$\xi_{t}=\mathbb{E}_{\mathbb{Q}_{t}^{\mu^{X}}}\left[-e^{-\int_{t}^{T}\beta_{s}ds}\,(Y-X)\Big{|}\mathcal{F}_{t}\right]=\mathbb{E}_{\lambda_{t}^{X}}\left[-(Y-X)\Big{|}\mathcal{F}_{t}\right],$

where $\lambda_{t}^{X}\triangleq
e^{-\int_{t}^{T}{\beta}^{X}_{s}ds}\,\mathbb{Q}^{{\mu}^{X}}_{t}$.

By applying the Comparison Theorem for BSDEs (see Theorem 7.1 in [20]) and by
the arguments above, we have:

$\rho_{t}(Y)-\rho_{t}(X)\geq\xi_{t}=\mathbb{E}_{\lambda_{t}^{X}}\left[-(Y-X)\Big{|}\mathcal{F}_{t}\right],\quad\mbox{
a.s. }\forall t\in[0,T].$

Hence $\lambda_{t}^{X}$ belongs to $\partial\rho_{t}(X)$. ∎

In the previous result we have shown that, given a cash-subadditive risk
measure $\rho_{t}$ induced by a BSDE, there exists a natural way to define
$\Lambda_{t}^{sub}$ as solution to a BSVIE. Once fixed an optimal scenario
$(D_{t}^{\beta},\mathbb{Q}_{t}^{\mu})$, the one-to-one correspondence between
$\rho_{t}$ and $\Lambda^{sub}_{t}$ suggests that we can build several
different CARs induced by BSVIEs considering the dynamics of the
subdifferential CAR instead of using the underlying risk measure. More in
general, following the strategy adopted in [33], we can define and build a CAR
$\Lambda_{t}$ as the first component of the solution to the BSVIE:

$\Lambda_{t}(X,Y)=-D_{t}X+\int_{t}^{T}g_{\Lambda}(t,s,\rho_{s}(Y),Z_{s}^{\rho_{s}(Y)},Z^{\Lambda^{s}(Y,Y)}(t,s),Z^{\Lambda}(t,s))ds-\int_{t}^{T}Z^{\Lambda}(t,s)dB_{s},$
(33)

where the driver
$g_{\Lambda}(\cdot,\cdot,\rho,Z^{\rho},Z^{\Lambda^{s}},\cdot)$ satisfies the
assumptions in Theorem 7 plus the further condition

$g_{\Lambda}(t,s,\rho,Z^{\rho},Z^{\Lambda^{s}},Z^{\Lambda^{s}})=g_{\Lambda^{s}}(t,s,\rho,Z^{\rho},Z^{\Lambda^{s}})\quad\mbox{
for any
}(t,s)\in\Delta,\rho\in\mathbb{R},Z^{\rho},Z^{\Lambda^{s}}\in\mathbb{R}^{n}.$
(34)

We note that the hypothesis (34) guarantees that
$\Lambda_{t}(X,X)=\Lambda^{sub}_{t}(X,X)=\rho_{t}(X)$ for every $X\in
L^{\infty}(\mathcal{F}_{T})$ and $t\in[0,T]$, implying that $\Lambda_{t}$ is a
CAR. Indeed, according to the uniqueness of the solution for a BSVIE and
taking
$(\Lambda_{t}(X,X),Z^{\Lambda}(t,s))=(\Lambda^{sub}_{t}(X,X),Z^{\Lambda^{s}}(t,s))$,
it holds that:

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

We stress that, while $\Lambda_{t}(X,X)$, $\Lambda_{t}^{sub}(X,X)$ and
$\rho_{t}(X)$ coincide a.s. $\forall t\in[0,T]$, the same does not necessarily
holds for $Z^{\Lambda^{s}(X,X)}\equiv Z^{\Lambda(X,X)}$ and $Z^{\rho(X)}$.

We provide here below some sufficient conditions on the driver $g_{\Lambda}$
under which the induced CAR satisfies some further properties.

######  Proposition 13.

Let
$g_{\Lambda}:\Omega\times\Delta\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$
be a driver satisfying the assumptions in Theorem 7 and condition (34) and let
$\Lambda$ be defined as in (33).

i) If $X\geq Y$, with $X,Y\in L^{\infty}(\mathcal{F}_{T})$, then
$\Lambda_{t}(X,Z)\leq\Lambda_{t}(Y,Z)$ for any $Z\in
L^{\infty}(\mathcal{F}_{T})$ and $t\in[0,T]$.

ii) $\Lambda_{t}(X+m_{t},Y)\geq\Lambda_{t}(X,Y)-m_{t}$ for any $X,Y\in
L^{\infty}(\mathcal{F}_{T}),m_{t}\in
L^{\infty}_{+}(\mathcal{F}_{t}),t\in[0,T]$.

iii) If $g_{\Lambda}(t,s,\rho,Z^{\rho},Z^{\Lambda^{s}},0)=0$ for any
$(t,s)\in\Delta,\rho\in\mathbb{R},Z^{\rho},Z^{\Lambda^{s}}\in\mathbb{R}^{n}$,
then $\Lambda_{t}(0,Y)=0$ for any $t\in[0,T],Y\in
L^{\infty}(\mathcal{F}_{T})$.

iv) If $g_{\Lambda}(t,s,\rho,Z^{\rho},Z^{\Lambda^{s}},Z^{\Lambda})\leq
g_{\Lambda^{s}}(t,s,\rho,Z^{\rho},Z^{\Lambda})$ for any
$\rho\in\mathbb{R},Z^{\rho},Z^{\Lambda^{s}},Z^{\Lambda}\in\mathbb{R}^{n},(t,s)\in\Delta$,
then $\Lambda_{t}$ satisfies no-undercut.

v) If
$\sum_{i=1}^{n}g_{\Lambda}(t,s,\rho,Z^{\rho},Z^{\Lambda^{s}},Z^{\Lambda^{i}})\leq
g_{\Lambda}(t,s,\rho,Z^{\rho},Z^{\Lambda^{s}},\sum_{i=1}^{n}Z^{\Lambda^{i}})$
for any $(t,s)\in\Delta,\rho\in\mathbb{R}$,
$Z^{\rho},Z^{\Lambda^{s}},Z^{\Lambda^{i}}\in\mathbb{R}^{n}$ (for
$i=1,\dots,n$), then $\Lambda_{t}$ satisfies sub-allocation for any
$t\in[0,T]$.

vi) If $g_{\Lambda}$ is convex in $Z^{\Lambda}$ for any
$(t,s)\in\Delta,\rho\in\mathbb{R},,Z^{\rho},Z^{\Lambda^{s}}\in\mathbb{R}^{n}$,
then $\Lambda_{t}$ satisfies weak-convexity.

###### Proof.

i) If $X\geq Y$ then $-D_{t}X\leq-D_{t}Y$ are elements of
$L^{\infty}([0,T],\mathcal{F}_{T};\mathbb{R})$ with continuous paths. The
thesis then follows by applying the Comparison Theorem (see Proposition 8).

ii), v), vi) can be checked easily by applying the Comparison Theorem.

iii) follows immediately by $\Lambda_{t}(0,Y)\equiv 0$ and
$Z^{\Lambda}(t,s)\equiv 0$ and by the uniqueness of the solution.

iv) For any $X,Y\in L^{\infty}(\mathcal{F}_{T})$

$\displaystyle\Lambda_{t}(X,Y)$ $\displaystyle=$ $\displaystyle-
D_{t}X+\int_{t}^{T}g_{\Lambda}(t,s,\rho_{s}(Y),Z_{s}^{\rho(Y)},Z^{\Lambda^{s}(Y,Y)}(t,s),Z^{\Lambda}(t,s))ds-\int_{t}^{T}Z^{\Lambda}(t,s)dB_{s}$
$\displaystyle\leq$ $\displaystyle-
D_{t}X+\int_{t}^{T}g_{\Lambda^{s}}(t,s,\rho_{s}(Y),Z_{s}^{\rho(Y)},Z^{\Lambda}(t,s))ds-\int_{t}^{T}Z^{\Lambda}(t,s)dB_{s}$
$\displaystyle=$ $\displaystyle\Lambda^{sub}_{t}(X,Y)\leq\rho_{t}(X)\ \mbox{
a.s. }\forall t\in[0,T],$

where the first inequality is given by the Comparison Theorem, the second
equality is due to the uniqueness of the solution, while the last inequality
follows from Proposition 12 (no-undercut of $\Lambda_{t}^{sub}$). ∎

We focus now on two relevant capital allocation rules: the gradient and the
Aumann-Shapley, generalized to our context.

####  3.2.1 Gradient capital allocation

Let $\rho_{t}:L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$ be a
dynamic cash-subadditive convex risk measure induced by the BSDE (27), where
the driver $g$ is convex in $(y,z)$, decreasing in $y$, and satisfies the
standard assumptions. Assume also that $g(t,y,z)$ is continuously
differentiable w.r.t. $(y,z)\in\mathbb{R}\times\mathbb{R}^{n}$. Moreover, for
$X,Y\in L^{\infty}(\mathcal{F}_{T})$, the dynamic gradient CAR is defined as

$\Lambda^{grad}_{t}(X,Y)\triangleq\lim_{h\to
0}\frac{\rho_{t}(X+hY)-\rho_{t}(X)}{h}=\nabla\rho_{t}(X;Y).$

Note that, under the assumptions above, the existence of the directional
derivative of $\rho_{t}$ at $X$ in every direction is guaranteed by Theorem
2.1 in [2].

######  Proposition 14.

Under the assumptions above on $\rho_{t}$ and on $g$,
$\Lambda_{t}^{grad}(X,Y)$ is the first component of the solution to the
following BSVIE:

$\Lambda^{grad}_{t}(X,Y)=-Xe^{-\int_{t}^{T}\beta_{u}du}-\int_{t}^{T}\mu_{s}Z^{grad}(t,s)\,ds-\int_{t}^{T}Z^{grad}(t,s)\,dB_{s},$
(35)

where
$\beta_{t}\triangleq-\partial_{\rho}g\left(t,\rho_{t}(Y),Z^{\rho(Y)}_{t}\right)$
and $\mu_{t}=-\nabla_{z}g\left(t,\rho_{t}(Y),Z_{t}^{\rho(Y)}\right)$.

###### Proof.

It is easy to check that all the hypotheses of Theorem 2.1 of [2] are
fulfilled. Then $\Lambda^{grad}_{t}(X,Y)$ satisfies

$\displaystyle\Lambda^{grad}_{t}(X,Y)$ $\displaystyle=$
$\displaystyle-X-\int_{t}^{T}\left(\beta_{s}\Lambda^{grad}_{s}(X,Y)-\mu_{s}Z^{\nabla}_{s}\right)\,ds-\int_{t}^{T}Z^{\nabla}_{s}\,dB_{s}$
$\displaystyle=$
$\displaystyle-X-\int_{t}^{T}\beta_{s}\Lambda^{grad}_{s}(X,Y)ds-\int_{t}^{T}Z^{\nabla}_{s}dB^{\mu}_{s},$

where $dB^{\mu}_{t}=dB_{t}+\mu_{t}dt$. By Proposition 2.2 of [19],
$\Lambda^{grad}$ can be rewritten as

$\Lambda^{grad}_{t}(X,Y)=\mathbb{E}_{\mathbb{Q}_{t}^{\mu}}[e^{-\int_{t}^{T}\beta_{u}du}\,(-X)|\mathcal{F}_{t}],$

where
$\mathbb{E}_{\mathbb{P}}[\frac{d\mathbb{Q}^{\mu}}{d\mathbb{P}}|\mathcal{F}_{t}]=\exp(\mu\cdot
B)_{t}$. Finally, by the Martingale Representation Theorem, for any fixed
$t\in[0,T]$ there exists a predictable process
$Z^{\operatorname{grad}}(t,\cdot)$ such that:

$\Lambda^{grad}_{t}(X,Y)=-Xe^{-\int_{t}^{T}\beta_{u}du}-\int_{t}^{T}Z^{grad}(t,s)\,dB^{\mu}_{s},$

hence the thesis. ∎

Note that the proof of Proposition 14 suggests a representation for the
gradient CAR $\Lambda_{t}^{grad}$ as in (35), which is similar to what we have
obtained in Example 11. It is also worth emphasizing that, if $g$ is
decreasing in $y$, sublinear, and continuously differentiable w.r.t. $(y,z)$,
then $\Lambda^{sub}_{t}(X,Y)\equiv\Lambda^{grad}_{t}(X,Y)$ for any
$t\in[0,T]$, by the uniqueness of the solution of the BSVIE and by Proposition
12.

####  3.2.2 Aumann–Shapley capital allocation

We study now the Aumann-Shapley capital allocation in the present framework.
We recall that the relevance and popularity of the Aumann-Shapley CAR is
mainly due to its relation with the Aumann-Shapley value and to the connection
with cooperative game theory. See [10, 15, 41] and the references therein for
details.

Consider a dynamic cash-subadditive risk measure $(\rho_{t})_{t\in[0,T]}$
driven by the BSDE (27). We can define $(\omega\times\omega)$ the following
functions:

$\displaystyle\Lambda^{AS}_{t}(X,Y)\triangleq\int_{0}^{1}\mathbb{E}_{\mathbb{Q}_{t}^{\mu^{\gamma
Y}}}\left[\left.-D^{\beta^{\gamma
Y}}_{t}X\right|\mathcal{F}_{t}\right]d\gamma$ (36)
$\displaystyle\Lambda^{p-AS}_{t}(X,Y)\triangleq\int_{0}^{1}\Lambda_{t}^{sub}(X,\gamma
Y)d\gamma.$ (37)

Note that the two formulations above coincide when the penalty function
$c_{t}$ is null for any $t\in[0,T]$ (e.g. for sublinear risk measures). We
refer to $\Lambda_{t}^{AS}$ as Aumann-Shapley capital allocation, while to
$\Lambda^{p-AS}_{t}$ as penalized Aumann-Shapley CAR. These capital allocation
rules have already been investigated for both the static and the dynamic case
when the underlying risk measure satisfies cash-additivity (see, for example,
[10, 28, 33]).

######  Theorem 15.

Let $\rho_{t}:L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$ be a
dynamic cash-subadditive risk measure induced by the BSDE (27) where $g$ is
convex in $(y,z)$, decreasing in $y$ and satisfies the standard assumptions.
Then $\Lambda_{t}^{AS}:L^{\infty}(\mathcal{F}_{T})\times
L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$ is an $1$-cash-
subadditive, monotone, normalized, full allocating CAR, while
$\Lambda_{t}^{p-AS}:L^{\infty}(\mathcal{F}_{T})\times
L^{\infty}(\mathcal{F}_{T})\to L^{\infty}(\mathcal{F}_{t})$ is an audacious
CAR satisfying no-undercut.

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

$\Lambda_{t}^{AS}(X,Y)=\mathbb{E}_{\mathbb{P}}\left[-\hat{L}^{Y}(T,t)X\Big{|}\mathcal{F}_{t}\right]$

with

$\hat{L}^{Y}(T,t)\triangleq\int_{0}^{1}L^{\gamma
Y}(T,t)d\gamma=\int_{0}^{1}\exp\left(-\frac{1}{2}\int_{t}^{T}|\mu_{s}^{\gamma
Y}|^{2}ds-\int_{t}^{T}\beta_{s}^{\gamma Y}ds-\int_{t}^{T}\mu_{s}^{\gamma
Y}dB_{s}\right)d\gamma,$

where, for any fixed $\gamma\in\mathbb{R}$ and $Y\in
L^{\infty}(\mathcal{F}_{T})$, $(\beta_{t}^{\gamma Y},\mu_{t}^{\gamma Y})$ is
an optimal scenario in the representation (15) of $\rho_{t}(\gamma Y)$.

###### Proof.

The guideline of this proof is a straightforward adaptation of Corollary 4.1
in [28] or Proposition 10 in [33], extending those results to the cash-
subadditive case. Let us define $F:\mathbb{R}\to L^{\infty}(\mathcal{F}_{t})$
as $F(\gamma)=\rho_{t}(\gamma X)$ for any fixed $t\in[0,T]$ and $X\in
L^{\infty}(\mathcal{F}_{T})$. We know that if $(\beta^{H}_{t},\mu^{H}_{t})$ is
an optimal scenario then the sub-probability
$\lambda_{t}^{H}=D_{t}^{\beta^{H}}\mathbb{Q}_{t}^{H}$ is an element of
$\partial\rho_{t}(H)$ for any $H\in L^{\infty}(\mathcal{F}_{T})$. It holds
that

$\displaystyle F^{\prime}_{-}(\gamma)=\lim_{h\uparrow 0}\frac{\rho_{t}(\gamma
X+hX)-\rho_{t}(\gamma X)}{h}\leq\mathbb{E}_{\lambda_{t}^{\gamma
X}}[-X|\mathcal{F}_{t}],$ $\displaystyle
F^{\prime}_{+}(\gamma)=\lim_{h\downarrow 0}\frac{\rho_{t}(\gamma
X+hX)-\rho_{t}(\gamma X)}{h}\geq\mathbb{E}_{\lambda_{t}^{\gamma
X}}[-X|\mathcal{F}_{t}],$

hence

$\int_{0}^{1}F^{\prime}_{-}(\gamma)d\gamma\leq\int_{0}^{1}\mathbb{E}_{\lambda_{t}^{\gamma
X}}[-X|\mathcal{F}_{t}]d\gamma=\Lambda^{AS}_{t}(X,X)\leq\int_{0}^{1}F^{\prime}_{+}(\gamma)d\gamma.$

By Proposition A.4 of [22], it holds $F^{\prime}_{-}=F^{\prime}_{+}$ a.s. and
the chain of equalities:

$\int_{0}^{1}F^{\prime}_{-}(\gamma)d\gamma=\Lambda_{t}^{AS}(X,X)=\int_{0}^{1}F^{\prime}_{+}(\gamma)d\gamma=F(1)-F(0)=\rho_{t}(X).$

Consequently, $\Lambda_{t}^{AS}$ is a CAR. From Remark 7.6 in [20], we know
that

$\frac{d\lambda_{t}^{\gamma
Y}}{d\mathbb{P}}=\exp\left(-\frac{1}{2}\int_{t}^{T}|\mu_{s}^{\gamma
Y}|^{2}ds-\int_{t}^{T}\beta_{s}^{\gamma Y}ds-\int_{t}^{T}\mu_{s}^{\gamma
Y}dB_{s}\right)\triangleq L^{\gamma Y}(T,t)$

then, by Fubini Theorem,

$\Lambda^{AS}_{t}(X,Y)=\int_{0}^{1}\mathbb{E}_{\lambda^{\gamma
Y}}[-X|\mathcal{F}_{t}]d\gamma=\int_{0}^{1}\mathbb{E}_{\mathbb{P}}[-XL^{\gamma
Y}(T,t)|\mathcal{F}_{t}]d\gamma=\mathbb{E}_{\mathbb{P}}\left[-X\int_{0}^{1}L^{\gamma
Y}(T,t)d\gamma\Big{|}\mathcal{F}_{t}\right],$

hence the thesis.

Monotonicity, normalization, and full allocation are due to standard
properties of conditional expectation, while $1$-cash-subadditivity of
$\Lambda^{AS}$ can be checked directly. For any $X,Y\in
L^{\infty}(\mathcal{F}_{T})$ and $m_{t}\in L^{\infty}(\mathcal{F}_{t})$,
indeed,

$\displaystyle\Lambda^{AS}_{t}(X+m_{t},Y)$ $\displaystyle=$
$\displaystyle\mathbb{E}_{\mathbb{P}}\left[-(X+m_{t})\int_{0}^{1}L^{\gamma
Y}(T,t)d\gamma\Big{|}\mathcal{F}_{t}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}_{\mathbb{P}}\left[-X\int_{0}^{1}L^{\gamma
Y}(T,t)d\gamma\Big{|}\mathcal{F}_{t}\right]+\mathbb{E}_{\mathbb{P}}\left[-m_{t}\int_{0}^{1}L^{\gamma
Y}(T,t)d\gamma\Big{|}\mathcal{F}_{t}\right]$ $\displaystyle=$
$\displaystyle\Lambda_{t}^{AS}(X,Y)-m_{t}\int_{0}^{1}\mathbb{E}_{\mathbb{Q}_{t}^{\mu^{\gamma
Y}}}\left[\exp\left(-\int_{t}^{T}\beta_{s}^{\gamma
Y}ds\right)\Bigg{|}\mathcal{F}_{t}\right]d\gamma$ $\displaystyle\geq$
$\displaystyle\Lambda_{t}^{AS}(X,Y)-m_{t},$

where the last inequality follows from $\beta^{\gamma Y}_{t}\geq 0$ for any
$\gamma\in[0,1]$.

It remains to prove the part concerning $\Lambda^{p-AS}_{t}(X,Y)$. It holds
that

$\displaystyle\Lambda^{p-AS}_{t}(X,X)=\Lambda_{t}^{AS}(X,X)-\int_{0}^{1}c_{t}(D_{t}^{\beta^{\gamma
X}}\mathbb{Q}_{t}^{\mu^{\gamma
X}})d\gamma=\rho_{t}(X)-\int_{0}^{1}c_{t}(D^{\beta^{\gamma
X}}_{t}\mathbb{Q}^{\mu^{\gamma X}}_{t})d\gamma,$ (38)

where the penalty term $c_{t}(D^{\beta^{\gamma X}}_{t}\mathbb{Q}^{\mu^{\gamma
X}}_{t})$ is given in (5). Then $\Lambda^{AS}_{t}(X,Y)\leq\rho_{t}(X)$, i.e.
it is an audacious CAR. Furthermore, the no-undercut property follows from

$\Lambda^{p-AS}_{t}(X,Y)=\int_{0}^{1}\left(\mathbb{E}_{\lambda_{t}^{\gamma
Y}}[-X|\mathcal{F}_{t}]-c_{t}(\lambda_{t}^{\gamma
Y})\right)\,d\gamma\leq\int_{0}^{1}\rho_{t}(X)d\gamma=\rho_{t}(X).$

∎

##  4 Further examples

In this section, we provide some examples of CARs driven by BSVIEs with
different drivers. More precisely, we show that the marginal capital
allocation and a generalized marginal capital allocation can be obtained via
the approach in (33). A further example will deal with the here defined cash-
subadditive entropic risk measure (CSERM) that allows to “build” different
CARs reflecting investor’s beliefs.

###  4.1 Dynamic marginal capital allocation rule

Let $\rho_{t}$ be a dynamic cash-subadditive risk measure driven by a BSDE.
Consider the dynamic marginal capital allocation rule, defined as

$\Lambda^{M}_{t}(X,Y)\triangleq\rho_{t}(Y)-\rho_{t}(Y-X),\quad X,Y\in
L^{\infty}(\mathcal{F}_{T}).$

Financially speaking, such an allocation quantifies the contribution of sub-
portfolio $X$ to the riskiness of the overall position $Y$. See Tasche [39] in
the static case for details.

The dynamic marginal capital allocation can be obtained through the
construction in (33) by means of a BSVIE. Indeed,

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

Defining $Z^{\Lambda^{M}}(t,s)\triangleq
Z^{\Lambda^{s}(Y,Y)}(t,s)-Z^{\Lambda^{s}(Y-X,Y-X)}(t,s)$ and

$\displaystyle
g_{\Lambda^{M}}(t,s,\rho_{s}(Y),Z^{\rho(Y)}_{s},Z^{\Lambda^{s}(Y)}(t,s),Z^{\Lambda^{M}}(t,s))$
$\displaystyle\triangleq
g_{\Lambda^{s}}(t,s,\rho_{s}(Y),Z^{\rho(Y)}_{s},Z^{\Lambda^{s}(Y,Y)}(t,s))-g_{\Lambda^{s}}(t,s,\rho_{s}(Y-X),Z_{s}^{\rho(Y-X)},Z^{\Lambda^{s}(Y,Y)}(t,s)-Z^{\Lambda^{M}_{s}}(t,s)),$

then $g_{\Lambda^{M}}$ satisfies condition (34) when
$g_{\Lambda^{s}}(t,s,\rho,Z^{\rho},0)=0$ for any $(t,s)\in\Delta$,
$(\rho,Z^{\rho})\in\mathbb{R}\times\mathbb{R}^{n}$ (true, for instance, when
$\rho_{t}$ is positively homogeneous). It then follows that the approach in
(33) covers also the case of the dynamic marginal CAR.

###  4.2 Generalized marginal dynamic CARs

Let us consider again a cash-subadditive risk measure $\rho_{t}$ induced via a
BSDE. Starting from the classical marginal capital allocation rule seen
before, we now define a dynamic generalized marginal capital allocation rule.

Given a stochastic adapted $\mathbb{R}^{n}$-valued process
$(\lambda_{t})_{t\in[0,T]}\in\mbox{BMO}(\mathbb{P})$ (see Section 7.1.2 [4]
for further details), we now consider a dynamic CAR $\Lambda$ with the
following driver

$\displaystyle
g_{\Lambda}(t,s,\rho_{s}(Y),Z^{\rho(Y)}_{s},Z^{\Lambda^{s}(Y,Y)}(t,s),Z^{\Lambda}(t,s))$
$\displaystyle=g_{\Lambda^{s}}(t,s,\rho_{s}(Y),Z^{\rho(Y)}_{s},Z^{\Lambda^{s}(Y,Y)}(t,s))+\lambda_{s}(Z^{\Lambda}(t,s)-Z^{\Lambda^{s}(Y,Y)}(t,s)).$

In such a case, condition (34) is automatically satisfied.

As shown in a while, the corresponding $\Lambda$ can be interpreted as a
dynamic generalized marginal capital allocation since

$\Lambda_{t}(X,Y)=\rho_{t}(Y)-\rho^{\lambda}_{t}(Y-X),$

where $\rho^{\lambda}$ is a dynamic risk measure that will defined here below
and depending on $(\lambda_{t})_{t\in[0,T]}$.

Evaluating the difference between $\Lambda_{t}(X,Y)$ and $\rho_{t}(Y)$ we get:

$\displaystyle\Lambda_{t}(X,Y)-\rho_{t}(Y)=\Lambda_{t}(X,Y)-\Lambda_{t}^{sub}(Y,Y)$
$\displaystyle=-D_{t}(X-Y)+\int_{t}^{T}\lambda_{s}(Z^{\Lambda}(t,s)-Z^{\Lambda^{s}(Y,Y)}(t,s))\,ds-\int_{t}^{T}\left(Z^{\Lambda}(t,s)-Z^{\Lambda^{s}(Y,Y)}(t,s)\right)\,dB_{s}$
$\displaystyle=-D_{t}(X-Y)-\int_{t}^{T}\left(Z^{\Lambda}(t,s)-Z^{\Lambda^{s}}(t,s)\right)\,dB^{\lambda}_{s},$

where $dB^{\lambda}_{t}\triangleq dB_{t}-\lambda_{t}dt$ is a Brownian motion
w.r.t. $(\mathcal{F}_{t})_{t\in[0,T]}$ because
$(\lambda_{t})_{t\in[0,T]}\in\mbox{BMO}(\mathbb{P})$, while
$D_{t}=D_{t}^{Y}=\exp\\{-\int_{t}^{T}\beta_{s}^{Y}ds\\}$ where $\beta^{Y}_{t}$
is the first component of the optimal scenario in the representation of
$\rho_{t}(Y)$. Taking the conditional expectation in the previous expression,
it holds that

$\Lambda_{t}(X,Y)-\rho_{t}(Y)=\mathbb{E}_{\mathbb{Q}^{\lambda}}[-D_{t}(Y-X)|\mathcal{F}_{t}],$

where
$\frac{d\mathbb{Q}^{\lambda}}{d\mathbb{P}}\triangleq\mathcal{E}(\int_{0}^{\cdot}\lambda_{t}dB_{t})_{T}$
is the measure corresponding to $B^{\lambda}$, defined via the stochastic
exponential. $\mathbb{E}_{\mathbb{Q}^{\lambda}}[-D_{t}(Y-X)|\mathcal{F}_{t}]$
turns out to be the component $\rho^{\lambda}_{t}(Y-X)$ of the (unique)
solution to the linear BSDE:

$\displaystyle\rho^{\lambda}_{t}(Y-X)$ $\displaystyle=$
$\displaystyle-(Y-X)+\int_{t}^{T}\left(-\beta^{Y}_{s}\rho_{s}^{\lambda}+\lambda_{s}Z^{\rho^{\lambda}}_{s}\right)ds-\int_{t}^{T}Z_{s}^{\rho^{\lambda}}\,dB_{s}$
$\displaystyle=$
$\displaystyle-(Y-X)+\int_{t}^{T}-\beta^{Y}_{s}\rho_{s}^{\lambda}ds-\int_{t}^{T}Z^{\rho^{\lambda}}_{s}\,dB^{\lambda}_{s}.$

$\rho^{\lambda}_{t}$ is then a cash-subadditive risk measure because the
driver
$g_{\rho^{\lambda}}(t,\rho^{\lambda},Z^{\rho^{\lambda}})=-\beta^{Y}_{t}\rho^{\lambda}+\lambda_{t}Z^{\rho^{\lambda}}$
is convex (or, better, linear) in $(\rho^{\lambda},Z^{\rho^{\lambda}})$ and
non-increasing in $\rho^{\lambda}$. It then follows that

$\Lambda_{t}(X,Y)=\rho_{t}(Y)-\rho^{\lambda}_{t}(Y-X).$

Such a $\Lambda$ can be therefore interpreted as a generalized marginal CAR,
being the difference between the risk associated to the total portfolio $Y$
through the underlying risk measure $\rho_{t}$ and the marginal contribution
of the sub-portfolio $(Y-X)$ evaluated via a different cash-subadditive risk
measure $\rho_{t}^{\lambda}$. Furthermore, $\rho_{t}^{\lambda}$ can be
understood as the degree of confidence- modulated by $\lambda_{s}$\- of the
investor in the market numéraire (see Section 6.2 in Barrieu and El Karoui
[4]) when the agent also faces with an ambiguous interest rate
$\beta^{Y}_{t}$.

###  4.3 Cash-subadditive entropic risk measures and capital allocations

In this section, we extend the classic entropic risk measure (solving a BSDE
with driver $g_{\rho^{E}}(t,Z^{\rho^{E}})=\frac{|Z^{\rho^{E}}|^{2}}{2\gamma}$
where $\gamma>0$ is the risk aversion of the agent) to the cash-subadditive
setting. Inspired by the work [20], we consider a driver whose local behaviour
is:

$\mathbb{E}_{\mathbb{P}}[-d\rho^{SE}_{t}|\mathcal{F}_{t}]=g_{\rho^{SE}}\,dt\triangleq\left(-\beta_{t}\rho^{SE}+\frac{|Z^{\rho^{SE}}|^{2}}{2\gamma}\right)dt,$

where $\beta_{t}$ is a positive, bounded and adapted process with the usual
interpretation as an ambiguous discount rate. In particular, this risk measure
is locally compatible with an agent with a certain risk aversion coefficient
$\gamma>0$ and adverse to interest rate ambiguity (see [4, 20] for a thorough
discussion). Clearly, when no ambiguity is possible, $\beta_{t}\equiv 0$ and
we regain the classic entropic risk measure. It is evident that
$g_{\rho^{SE}}$ does not satisfy a Lipschitz condition, being quadratic in
$Z^{\rho^{SE}}$. Nevertheless, the theory we have developed is still robust.
Indeed, representation (15) holds also for risk measures with quadratic growth
(see Theorem 7.5 in [20]). Moreover, $g_{\rho^{SE}}$ is differentiable both in
$\rho^{SE}$ and $Z^{\rho^{SE}}$, hence

$\displaystyle g_{\Lambda^{s}}(t,s,\rho^{SE},Z^{\rho^{SE}},Z^{\Lambda^{s}})$
$\displaystyle=D_{t,s}\left[-g_{\rho^{SE}}(s,\rho^{SE}_{s},Z^{\rho^{SE}}_{s})-\beta_{s}\rho_{s}^{SE}+\frac{|Z^{\rho^{SE}}|^{2}}{\gamma}\right]-\frac{1}{\gamma}Z^{\rho^{SE}}_{s}Z^{\Lambda^{s}}(t,s)$
$\displaystyle=\frac{D_{t,s}}{2\gamma}|Z^{\rho^{SE}}_{s}|^{2}-\frac{1}{\gamma}Z^{\rho^{SE}}_{s}Z^{\Lambda^{s}}(t,s),$

because of the relation between gradient and subdifferential:

$\displaystyle\partial^{sub}_{\rho}g(t,\rho^{SE}_{t},Z^{\rho^{SE}}_{t})=-\partial_{\rho}g(t,\rho_{t}^{SE},Z^{\rho^{SE}}_{t})=\beta_{t},$
$\displaystyle\partial^{sub}_{z}g(t,\rho^{SE}_{t},Z^{\rho^{SE}}_{t})=-\partial_{z}g(t,\rho_{t}^{SE},Z^{\rho^{SE}}_{t})=-\frac{Z^{\rho^{SE}}_{t}}{\gamma}.$

Although it is not clear if the BSVIE with driver
$g_{\Lambda^{s}}(t,s,\rho^{SE},Z^{\rho^{SE}},Z^{\Lambda^{s}})$ admits a unique
solution (since the Lipschitz coefficient $Z^{\rho^{SE}}_{s}$ is no longer
bounded and
$g(t,s,\rho^{SE},Z^{\rho^{SE}},0)=\frac{D_{t,s}}{2\gamma}|Z^{\rho^{SE}}_{s}|^{2}$
is not square integrable), the existence and uniqueness of the solution is
proved in Lemma 16 here below.

We define now a generalized notion of entopic risk measure in the cash-
subadditive case, with interest rate $(\beta_{t})_{t\in[0,T]}$ (same as
before) and different risk aversion parameter $\gamma_{1}>0$, i.e.

$g_{\rho^{\gamma_{1}}}(t,\rho^{\gamma_{1}},Z^{\rho^{\gamma_{1}}})=-\beta_{t}\rho^{\gamma_{1}}+\frac{1}{2\gamma_{1}}|Z^{\rho^{\gamma_{1}}}|^{2}.$

The corresponding dynamic risk measure is a cash-subadditive and convex risk
measure that will be called cash-subadditive entropic risk measure (CSERM).

Set now

$\displaystyle
g_{\Lambda^{SE}}(t,s,\rho^{SE}_{s}(Y),Z^{\rho^{SE}(Y)}_{s},Z^{\Lambda^{s}(Y,Y)}(t,s),Z^{\Lambda^{SE}}(t,s))$
$\displaystyle\triangleq
g_{\Lambda^{s}}(t,s,\rho^{SE}_{s}(Y),Z^{\rho^{SE}(Y)}_{s},Z^{\Lambda^{s}(Y,Y)}(t,s))+\frac{2}{\gamma_{1}}Z^{\rho^{\gamma_{1}}}_{s}(Z^{\Lambda^{SE}}(t,s)-Z^{\Lambda^{s}}(t,s)).$

Lemma 16 guarantees then that the BSVIE with driver $g_{\Lambda^{SE}}$ admits
a unique solution. As seen below, if a solution (with whatsoever regularity)
is provided then $\Lambda^{SE}_{t}$ is a bounded process. Following the same
steps as in the generalized marginal CARs case,

$\Lambda^{SE}_{t}(X,Y)=\rho^{SE}_{t}(Y)-\rho_{t}^{\nabla}(Y-X),$

where

$\rho_{t}^{\nabla}(Y-X)=-(Y-X)+\int_{t}^{T}\left[-\beta_{s}\rho_{s}^{\nabla}(Y-X)+\frac{2}{\gamma_{1}}Z_{s}^{\rho^{\gamma_{1}}}Z_{s}^{\rho^{\nabla}}\right]ds-\int_{t}^{T}Z^{\rho^{\nabla}}_{s}dB_{s}$

admits a unique bounded solution. Furthermore,

$\Lambda^{SE}_{t}(X,Y)=\rho_{t}^{SE}(Y)-\nabla\rho_{t}^{\gamma_{1}}(Y-X;Y),$

where $\nabla\rho_{t}^{\gamma_{1}}$ is the gradient capital allocation
considering $\rho_{t}^{\gamma_{1}}$ as underlying risk measure (see Kromer and
Overbeck [28] for further details). Indeed,

$\rho^{\nabla}_{t}(Y-X)=\nabla\rho^{\gamma_{1}}_{t}(Y-X;Y)\mbox{ a.s.}$ (39)

because, according to Theorem 2.1 in [2],

$\nabla\rho^{\gamma_{1}}_{t}(Y-X;Y)=-(Y-X)+\int_{t}^{T}\left[-\beta_{s}\nabla\rho^{\gamma_{1}}_{s}(Y-X;Y)+\frac{2}{\gamma_{1}}Z^{\rho^{\gamma_{1}}}_{s}\nabla
Z^{\rho^{\gamma_{1}}}_{s}\right]ds-\int_{t}^{T}\nabla
Z^{\rho^{\gamma_{1}}}_{s}dB_{s}$

and by uniqueness of the solution.

Summing up, the capital allocation rules associated to the driver
$g_{\Lambda^{SE}}$ can be expressed in terms of the difference between the
risk of the total portfolio $Y$ measured by the cash-subadditive entropic risk
measure with risk aversion $\gamma$ and the risk of the margin portfolio
$(Y-X)$ given by the Gateaux derivative of the CSERM with risk aversion
parameter $\gamma_{1}$.

Finally, we prove the result on the existence and uniqueness in a weaker sense
mentioned before.

######  Lemma 16.

In the previous setting, there exists a unique solution
$(\Lambda^{sub},Z^{s})\in\mathcal{H}^{1,\infty}$ to the BSVIE

$\Lambda^{sub}_{t}(X,Y)=-D_{t}X+\int_{t}^{T}\left[\frac{1}{2\gamma}D_{t,s}|Z^{\rho^{SE}}_{s}|^{2}-\frac{1}{\gamma}Z^{\rho^{SE}}_{s}Z^{\Lambda^{s}}(t,s)\right]ds-\int_{t}^{T}Z^{\Lambda^{s}}(t,s)dB_{s}.$
(40)

Moreover, the BSVIE

$\Lambda_{t}^{SE}(X,Y)=-D_{t}X+\int_{t}^{T}g_{\Lambda^{SE}}(t,s,\rho_{s}^{SE}(Y),Z^{\rho^{SE}}_{s},Z^{\Lambda^{s}(Y,Y)}(t,s),Z^{SE}(t,s))ds-\int_{t}^{T}Z^{SE}(t,s)dB_{s}$
(41)

admits a unique solution
$(\Lambda^{SE},Z^{\Lambda^{SE}})\in\mathcal{H}^{1,\infty}$ with the further
regularity $\Lambda^{SE}\in L^{\infty}((0,T),\mathbb{F}).$

###### Proof.

We start proving that there exists a unique solution to (40). We know that
$Z^{\rho^{SE}}_{t}\in\mbox{BMO}(\mathbb{P})$ (see [42]), then
$dB_{t}^{\rho^{SE}}=dB_{t}+\frac{1}{\gamma}Z^{\rho^{SE}}_{t}dt$ is a Brownian
motion w.r.t.
$\mathbb{Q}^{\rho^{SE}}=\mathcal{E}\left(\int_{0}^{\cdot}\frac{1}{\gamma}Z^{\rho^{SE}}_{t}dB_{t}\right)_{T}$.
Hence, equation (40) becomes

$\Lambda^{sub}_{t}(X,Y)=-D_{t}X+\int_{t}^{T}\frac{1}{2\gamma}D_{t,s}|Z^{\rho^{SE}}_{s}|^{2}ds-\int_{t}^{T}Z^{\Lambda^{s}}(t,s)dB^{\rho^{SE}}_{s}=\varphi(t)-\int_{t}^{T}Z^{\Lambda^{s}}(t,s)dB^{\rho^{SE}}_{s},$

with

$\varphi(t)=-D_{t}X+\int_{t}^{T}\frac{1}{2\gamma}D_{t,s}|Z^{\rho^{SE}}_{s}|^{2}ds.$

We observe that $\varphi\in L^{1}(\Omega,\mathcal{F}_{T};L^{\infty}(0,T))$,
given that $D_{t}\in L^{\infty}((0,T),\mathcal{F}_{T})$ and
$Z^{\rho^{SE}}\in\mbox{BMO}(\mathbb{P})$. The last equation enables us to
avoid dealing with stochastic Lipschitz coefficients. Let us consider the
family of BSDEs whose elements are defined for each fixed $(t,s)\in\Delta$ as
follows:

$\eta(s;t)=\varphi(t)-\int_{s}^{T}\zeta(r;t)dB_{r}^{\rho^{SE}},\ \
(t,s)\in\Delta.$

There exists a unique solution $(\eta(\cdot;t),\zeta(\cdot;t))\in
L^{1}(\Omega,\mathbb{F},L^{\infty}(t,T))\times
L^{1}(\Omega,\mathbb{F};L^{2}(t,T))$ for each $t\in[0,T]$, by Theorem 3 in
[21]. Moreover, we have:

$\displaystyle|\eta(s;t)|$
$\displaystyle\leq\mathbb{E}\left[|\varphi(t)|\big{|}\mathcal{F}_{s}\right]\leq
C\mathbb{E}\left[\left.|X|+\int_{t}^{T}|Z^{\rho^{SE}}_{r}|^{2}dr\right|\mathcal{F}_{s}\right]$
$\displaystyle\leq
C\left(\|X\|_{\infty}+\mathbb{E}\left[\int_{t}^{s}|Z^{\rho^{SE}}_{r}|^{2}dr\right]+\mathbb{E}\left[\left.\int_{s}^{T}|Z^{\rho^{SE}}_{r}|^{2}ds\right|\mathcal{F}_{s}\right]\right)$
$\displaystyle\leq
C\left(\|X\|_{\infty}+\mathbb{E}\left[\int_{t}^{T}|Z^{\rho^{SE}}_{r}|^{2}dr\right]+\sup_{m\in[t,T]}\mathbb{E}\left[\left.\int_{m}^{T}|Z^{\rho^{SE}}_{r}|^{2}dr\right|\mathcal{F}_{m}\right]\right)\leq
C,$

where $C>0$ is a constant that does not depend on $(t,s)\in\Delta$ and can
vary from line to line. Second inequality follows from the boundedness of
$D_{t}$ while last inequality is due to
$Z^{\rho^{SE}}\in\mbox{BMO}(\mathbb{P})$. Thus, we have:

$\mathbb{E}\left[\sup_{(t,s)\in\Delta}|\eta(s;t)|\right]\leq C<+\infty.$

By Proposition 1 in [21], for any $(t,s)\in\Delta$ we have the estimate:

$\mathbb{E}\left[\left(\int_{s}^{T}|\zeta(r;t)|^{2}dr\right)^{1/2}\right]\leq
C\mathbb{E}\left[\sup_{m\in[s,T]}|\eta(m;t)|\right],$

taking the sup over $t\in[0,T]$ we have:

$\sup_{t\in[0,T]}\mathbb{E}\left[\left(\int_{s}^{T}|\zeta(r;t)|^{2}dr\right)^{1/2}\right]\leq
C\sup_{t\in[0,T]}\mathbb{E}\left[\sup_{m\in[s,T]}|\eta(m;t)|\right]\leq
C^{\prime}<+\infty.$

Setting $\Lambda^{sub}(t)\equiv\eta(t;t)$ and
$Z^{\Lambda^{s}}(t,s)\equiv\zeta(s;t)$ with $(t,s)\in\Delta$, the thesis
follows (uniqueness is given by the previous estimates together with linearity
of the equation, by taking $\varphi\equiv 0$).

Now we are ready to show the existence and uniqueness for the BSVIE (41). We
are going to build the solution $(\Lambda^{SE}_{t},Z^{SE}(t,s))$ by means of
the solution to the BSVIE:

$\Lambda^{\prime}_{t}(X-Y)=-D_{t}(X-Y)-\int_{t}^{T}Z^{\Lambda^{\prime}}(t,s)dB^{\gamma_{1}}_{s},$

where $dB^{\gamma_{1}}_{s}\triangleq
dB_{s}-\frac{2}{\gamma_{1}}Z^{\gamma_{1}}_{s}ds$ is a
$\mathbb{Q}^{\gamma_{1}}$-Brownian motion given that
$\frac{2}{\gamma_{1}}Z^{\gamma_{1}}_{s}\in\mbox{ BMO}(\mathbb{P})$. The last
equation admits a unique bounded solution according to Theorem 7. Set now

$\displaystyle\Lambda^{SE}_{t}(X,Y)\triangleq\Lambda^{\prime}_{t}(X-Y)+\rho_{t}^{SE}(Y)=\Lambda^{\prime}_{t}(X-Y)+\Lambda^{sub}_{t}(Y,Y)\in
L^{\infty}([0,T],(\mathcal{F}_{t})_{t\in[0,T]};\mathbb{R})$ $\displaystyle
Z^{SE}(t,s)\triangleq Z^{\Lambda^{\prime}}(t,s)+Z^{\Lambda^{s}(Y,Y)}(t,s)\in
L^{\infty}((0,T);L^{1}(\Omega,(\mathcal{F}_{t})_{t\in[t,T]};L^{2}(t,T))).$

By construction, $(\Lambda^{SE}_{t}(X,Y),Z^{\Lambda^{SE}}(t,s))$ is a solution
of equation (41) with the above mentioned regularity. It remains to prove
uniqueness of the solution. Let us consider another solution
$(\Lambda^{1}_{t},Z^{\Lambda^{1}}(t,s))\in\mathcal{H}^{1,\infty}$ to equation
(41). Then

$\displaystyle\Lambda^{SE}_{t}(X,Y)-\Lambda^{1}_{t}(X,Y)$ $\displaystyle=$
$\displaystyle\int_{t}^{T}\frac{2}{\gamma_{1}}Z^{\rho^{\gamma_{1}}}_{t}(Z^{\Lambda^{SE}}(t,s)-Z^{\Lambda^{1}}(t,s))\,ds-\int_{t}^{T}(Z^{\Lambda^{SE}}(t,s)-Z^{\Lambda^{1}}(t,s))\,dB_{s}$
$\displaystyle=$
$\displaystyle\int_{t}^{T}(Z^{\Lambda^{SE}}(t,s)-Z^{\Lambda^{1}}(t,s))\,dB^{\gamma_{1}}_{s},$

where $dB_{t}^{\gamma_{1}}\triangleq
dB_{t}-\frac{2}{\gamma_{1}}Z^{\rho^{\gamma_{1}}}_{t}dt$ is a
$\mathbb{Q}^{\gamma_{1}}$-Brownian motion. Since the last BSVIE admits a
unique solution given by $(\Lambda^{SE}_{t}-\Lambda^{1}_{t}\equiv
0,Z^{\Lambda^{SE}}(t,s)-Z^{\Lambda^{1}}(t,s)\equiv
0)\in\mathcal{H}^{1,\infty}$ (thanks to the previous results), the uniqueness
then follows. ∎

##  5 Conclusions

To sum up, in this paper we have focused on capital allocations associated to
cash-subadditive risk measures, adopting an axiomatic approach without
assuming Gateaux differentiability and providing examples where the underlying
risk measure is driven by a BSDE. To the best of our knowledge, this is the
first attempt to deal with dynamic capital allocation rules when the
underneath risk measure does not fulfil the axiom of cash-additivity. More
precisely, we have extended the results obtained in Mastrogiacomo and Rosazza
Gianin [31] and in Kromer and Overbeck [28] to the cash-subadditive case. At
the same time, differently from Centrone and Rosazza Gianin [10] dealing with
static CARs associated to convex and quasi-convex risk measures, we encompass
in our treatment also the dynamic setting.

For risk measures induced by BSDEs, [31, 28] showed that the corresponding
capital allocations follow still a BSDE; in particular, this result is due to
the axiom of cash-additivity. On the Volterra side, instead, Kromer and
Overbeck [30] proved that the dynamic gradient capital allocation of risk
measures generated by a BSVIE is still given by a BSVIE. Differently from
these approaches mapping either from BSDEs to BSDEs or from BSVIEs to BSVIEs
in the cash-additive case, here we have provided an alternative way to build
capital allocation rules starting from cash-subadditive risk measures induced
by BSDEs.

Compared to the existing literature, the main novelties of our work consist
of: firstly, weakening cash-additivity with cash-subadditivity at the level of
risk measures and evaluating the impact on capital allocations; secondly,
proving that the subdifferential CAR associated to a cash-subadditive risk
measure induced by a BSDE follows a BSVIE; thirdly, having introduced a
general way to build different CARs given by BSVIEs. This general approach
allows to treat the cash-subadditive case similarly as done for cash-additive
risk measures in [31], switching from probabilities to sub-probabilities and
from BSDEs to BSVIEs.

Acknowledgements: The authors are members of Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA), Italy. The authors
thank Ilaria Peri and the participants at the Quantitative Finance Workshop
2023 in Gaeta, Italy, for comments and discussions.

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