# Fully-dynamic risk measures: horizon risk, time-consistency, and relations
with BSDEs and BSVIEs

Giulia Di Nunno Department of Mathematics, University of Oslo, P.O. Box 1053
Blindern, N-0316 Oslo Norway. Email<EMAIL_ADDRESS>- Norwegian School
of Economics, Helleveien 30, N-5045 Bergen, Norway. Emanuela Rosazza Gianin
Department of Statistics and Quantitative Methods, University of Milano
Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano Italy. Email:
<EMAIL_ADDRESS>
(November 15, 2023)

###### Abstract

In a dynamic framework, we identify a new concept associated with the risk of
assessing the financial exposure by a measure that is not adequate to the
actual time horizon of the position. This will be called horizon risk. We
clarify that dynamic risk measures are subject to horizon risk, so we propose
to use the fully-dynamic version. To quantify horizon risk, we introduce
h-longevity as an indicator. We investigate these notions together with other
properties of risk measures as normalization, restriction property, and
different formulations of time-consistency. We also consider these concepts
for fully-dynamic risk measures generated by backward stochastic differential
equations (BSDEs), backward stochastic Volterra integral equations (BSVIEs),
and families of these. Within this study, we provide new results for BSVIEs
such as a converse comparison theorem and the dual representation of the
associated risk measures.

Keywords: Fully-dynamic risk measures, time-consistency, BSDEs, BSVIEs,
converse comparison theorem for BSVIEs, dual representation, horizon risk,
h-longevity

MSC2020: 60H10, 60H20, 91B70, 91G70

##  1 Introduction

Monetary risk assessment spans across time horizons with different length,
from the very short ones for trading operations to the decades-long ones
typical of sovereign wealth or pension funds. Hence the use of adequate risk
measures across time as well as a time-consistent evaluation of risk are
important factors in risk quantification and management.

We address the problem of using an appropriate risk evaluation for the given
time horizon. To explain, it is not correct to use a risk measure designed for
long term positions to evaluate risks that occur in the short term. This is
considered a “common error” in risk quantification by some quants, see e.g.
[12]. Indeed, bearing in mind that the purpose of risk assessment and
management is not avoiding risk, but giving the rightful possibility to invest
in risky assets with a reasonable control on what is acceptable to the
investor, then it is clear that this form of “horizon risk” should be tackled.
This is particularly evident in life insurance with an impact on pension funds
and health insurance. For example, a wrong estimate of the individual
remaining lifetime brings to a wrong use of the associated mortality tables
and, consequently, of the risk of an underestimation of insurance premia and,
by domino effect, a wrong estimation of capital requirements and their
associated financial management. In this work, we identify horizon risk and we
propose a way to quantify this via the horizon longevity, or h-longevity in
short. For this we work with the general framework of fully-dynamic risk
measures that embeds the classical dynamic risk measures. Indeed, a fully-
dynamic risk measure is a family of risk measures
$(\rho_{su})_{s,u}\triangleq(\rho_{su})_{0\leq s\leq u\leq T}$ $(\text{with
}T<\infty)$ indexed by two time parameters, the first represents the
evaluation time, the second the horizon to which the risk pertains. Instead, a
dynamic risk measure is a family of risk measures
$(\rho_{s})_{s}\triangleq(\rho_{s})_{0\leq s\leq T}$ indexed by only one time
parameter representing the time of risk evaluation. So we have that

$\rho_{s}=\rho_{sT}.$ (1)

Whenever we consider a risk $X$ occurring at time $t\in[0,T]$ (hence
$\mathcal{F}_{t}$-measurable for some given information flow), we can consider
different risk evaluations at $s$, either $\rho_{st}(X)$ or $\rho_{su}(X)$,
for any other horizon $u>t$ (even much later than $t$), since
$\mathcal{F}_{t}\subseteq\mathcal{F}_{u}$. Horizon longevity penalizes the use
of the “wrong” risk measure, that is the one that does not pertain to the
right time horizon.

The impact of the length of the time horizon on risk evaluation has already
been detected in the work [16] on epistemic uncertainty. There, it comes out
that ambiguity on the model choice is growing with the time horizon. While
[16] is not related with our considerations, it shows anyhow that the time
horizon effects the precision on risk quantification.

Another concept under investigation in this paper is time-consistency, which
has different formulations and it plays an important role in dynamic risk
evaluation. See e.g. [1, 4, 6, 7].

In our study we highlight that both h-longevity and time-consistency are
linked to the restriction property (see [7]), that is

$\rho_{st}(X)=\rho_{su}(X),\qquad\text{for}\;\mathcal{F}_{t}\text{-measurable}\;X,\quad
u\geq t.$ (2)

Indeed, the presence of restriction induces that the different formulations of
time-consistency are equivalent, while the absence of restriction provides the
very possibility to introduce h-longevity. The restriction property is
naturally satisfied by dynamic risk measures $(\rho_{s})_{s}$ (see (1)) and
this may be a reason for which the horizon risk has not been earlier
identified in its own being and hence quantified. Similarly, also
normalization, i.e.

$\rho_{su}(0)=0,$ (3)

is discovered to play a crucial role in the relationships among the concepts
above. Note that normalization is often standardly assumed in many financial
risk evaluations. However, it is questionable in the evaluation of
environmental strategies, namely a non-action (zero investment) can carry a
positive risk. Note that, by ESG regulations, these positions and their
entailed risks have to be reported.

In the first part of this paper, we work in full generality with an axiomatic
setting. In the second part we concentrate on risk measures induced by
backward dynamics. We focus on the Brownian framework (see, e.g., [3, 18, 22,
25]) and use a BSDE with driver $g$ to generate a fully-dynamic risk measure
$(\rho_{su})_{s,u}$. Then we study how the properties of the driver are
connected to the concepts of time-consistency, h-longevity, restriction, and
normalization.

Since a fully-dynamic risk measure $(\rho_{su})_{s,u}$ depends on the horizons
$u\in[0,T]$, then we can also generate it from a family of BSDEs with drivers
$\mathcal{G}=(g_{u})_{u}$. In this way it is emphasized that each period
$[s,u]$ is associated to the BSDE with driver $g_{u}$ providing the risk
evaluation $\rho_{su}$. In this framework we investigate again the
relationships among the concepts of interest with the family of drivers. In
this context we prove a Comparison Theorem for BSDEs with different time
horizons useful for the study of h-longevity.

Recently, backward stochastic Volterra integral equations (BSVIEs) have been
suggested to generate dynamic risk measures, see [29, 2]. However, at that
stage, some important results on BSVIEs were still lacking. Here we provide a
dual representation of risk measures induced by BSVIEs and a converse
comparison theorem for BSVIEs. With these two results we have a full picture
of the relationship between the properties of the driver and those of the
corresponding risk measures as well as a comprehensive study of time-
consistency and horizon longevity in the Volterra setting.

We recall that a BSVIE is induced by a family of BSDEs, see [30]. This has
been a further motivation to study fully-dynamic risk measures induced by a
BSVIE with Volterra driver $g=g(t,s,\cdot)$, $s\geq t$. As a final stage we
have also considered families of BSVIEs with Volterra drivers
$\mathcal{G}=(g_{u})_{u}$ again to emphasize the role of the time horizon in
the risk evaluation.

Working with families of backward equations has made evident the surprisingly
crucial role of the restriction property in risk measurement across time. For
example, it turns out that the only way to have a strong (recursive) time-
consistency is to work with fully-dynamic risk measures induced by a single
standard BSDE.

Summarizing, in this work we have first identified horizon risk in its own
being and proposed one way to quantify it. We have hence considered fully-
dynamic risk measures and studied the different forms of time-consistency and
h-longevity, showing how the restriction property is actually playing a
crucial role in these matters. Our work has covered both the general axiomatic
approach and the risk measures generated by backward dynamics, both single and
in families, both of standard and of Volterra type. In this we have seen how
the properties of the drivers reflect the properties of the fully-dynamic risk
measures in respect of the risk evaluations across time. In between we have
also provided results on BSDE and BSVIEs of independent interest.

##  2 Fully-dynamic risk measures, time-consistency, and horizon risk

In the sequel, we will focus on fully-dynamic risk measures that have been
recently introduced by Bion-Nadal and Di Nunno [7]. See also [5], where this
concept was actually simply called dynamic risk measure.

###### Definition 1

A _fully-dynamic risk measure_ is a family
$(\rho_{st})_{s,t}=(\rho_{st})_{0\leq s\leq t\leq T}$ of risk measures indexed
by two time parameters

$\rho_{st}:L^{p}(\mathcal{F}_{t})\to L^{p}(\mathcal{F}_{s}),\mbox{ with
}p\in[1,+\infty],$

that for all $X,Y\in L^{p}(\mathcal{F}_{t})$ are:

  * -

monotone: $\rho_{st}(X)\leq\rho_{st}(Y)$ for $X\geq Y$,

  * -

convex: $\rho_{st}(\lambda
X+(1-\lambda)Y)\leq\lambda\rho_{st}(X)+(1-\lambda)\rho_{st}(Y)$ for all
$\lambda\in[0,1]$,

  * -

$\mathcal{F}_{s}$-translation invariant (cash-invariant):
$\rho_{st}(X+Y)=\rho_{st}(X)-Y$ for all $Y\in L^{p}(\mathcal{F}_{s})$

and, for $p=\infty$,

  * -

continuous from below: if $X_{n}\uparrow X$ as $n\to+\infty$, then
$\rho_{st}(X_{n})\to\rho_{st}(X)$ in $L^{p}(\mathcal{F}_{s})$, as
$n\to+\infty$.

In the definition above, continuity from below is assumed only in the case
where $p=\infty$. In fact, for any $p\in[1,\infty)$, it is implied by the
other assumptions, see [7, Remark 2.5]. Furthermore, fully-dynamic risk
measures satisfy weak $\mathcal{F}_{s}$-homogeneity, i.e.

$1_{A}\rho_{st}(X)=1_{A}\rho_{st}(1_{A}X),\quad\mbox{ for any }X\in
L^{p}(\mathcal{F}_{t}),A\in\mathcal{F}_{s}$

(see [7, Remark 2.6]) and have the following dual representations

$\displaystyle\rho_{st}(X)$ $\displaystyle=$
$\displaystyle\underset{Q\in\mathcal{Q}_{st}}{\operatorname{ess.\\!max}}\;\\{E_{Q}[-X|\mathcal{F}_{s}]-\alpha_{st}(Q)\\}$
(4) $\displaystyle=$ $\displaystyle\underset{\begin{subarray}{c}Q\ll P:\\\
E_{P}[\alpha_{st}(Q)]<\infty\end{subarray}}{\operatorname{ess.\\!max}}\\{E_{Q}[-X|\mathcal{F}_{s}]-\alpha_{st}(Q)\\}$
(5)

where

$\mathcal{Q}_{st}\triangleq\left\\{Q\mbox{ on }\mathcal{F}_{t}:Q\ll P\mbox{
and }Q|_{\mathcal{F}_{s}}\equiv P|_{\mathcal{F}_{s}}\right\\}$ (6)

and

$\alpha_{st}(Q)\triangleq\underset{X\in
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\;\\{E_{Q}[-X|\mathcal{F}_{s}]-\rho_{st}(X)\\}$
(7)

is the minimal penalty functional. See [7, Proposition 2.8] for details.

We stress that, in general, the risk measures in the family of the fully
dynamic risk measure are not normalized, i.e. $\rho_{st}(0)\neq 0$, and the
family does not satisfy the restriction property, that is
$\rho_{rt}(Y)\neq\rho_{rs}(Y)$ for $Y\in L^{p}(\mathcal{F}_{s})$, for $s\leq
t$.

In view of the dynamic nature of the risk evaluation, we consider the
relationships among the inter-temporal evaluations given by the fully-dynamic
risk measure. For this, we recall here below the notions of time-consistency
mostly used in the literature and their connections expressed in the present
setting. See, e.g., [1, 4, 6], among others.

###### Definition 2

  * •

_Strong time-consistency_ (or _recursivity_): for any $s,t,u\in[0,T]$ with
$s\leq t\leq u$,

$\rho_{st}(-\rho_{tu}(X))=\rho_{su}(X)\quad\mbox{ for any }X\in
L^{p}(\mathcal{F}_{u}).$ (8)

  * •

_Order time-consistency_ : for any $s,t,u\in[0,T]$ with $s\leq t\leq u$,

$\rho_{tu}(X)=\rho_{tu}(Y),\quad X,Y\in
L^{p}(\mathcal{F}_{u})\Longrightarrow\rho_{su}(X)=\rho_{su}(Y).$ (9)

  * •

_Weak time-consistency_ : for any $s,t,u\in[0,T]$ with $s\leq t\leq u$,

$\rho_{su}(\rho_{tu}(0)-\rho_{tu}(X))=\rho_{su}(X)\quad\mbox{ for any }X\in
L^{p}(\mathcal{F}_{u}).$ (10)

Observe that strong time-consistency is equivalent to the cocycle condition on
the minimal penalties:

$\alpha_{su}(Q)=\alpha_{st}(Q|_{\mathcal{F}_{t}})+E_{Q}[\alpha_{tu}(Q)|\mathcal{F}_{s}],\quad\forall
s,t,u,Q\in\mathcal{Q}_{su},$ (11)

together with the m-stability of the Radon-Nykodym derivatives associated to
the corresponding sets of measures $(\mathcal{Q}_{st})_{s,t}$, i.e. the
probability measure $S$ defined by

$\frac{dS}{dP}=\frac{dQ}{dP}\,\frac{dR}{dP},\quad\textrm{for all
}Q\in\mathcal{Q}_{st},R\in\mathcal{Q}_{tu}$ (12)

belongs to $\mathcal{Q}_{su}$. See [5, Theorem 2.5]. The operation in (12) is
called pasting of probability measures, also shortly denoted $Q\cdot R$.
Naturally, any measure $S\in\mathcal{Q}_{su}$ admits a representation in the
form (12). In fact, we have

$\frac{dS}{dP}=\frac{dS|_{\mathcal{F}_{t}}}{dP}\,\frac{dR}{dP}\quad\textrm{with
}\frac{dR}{dP}=\frac{\frac{dS}{dP}}{E_{P}\big{[}\frac{dS}{dP}|\mathcal{F}_{t}\big{]}}1_{A}+1_{A^{c}}$

where
$A\triangleq\big{\\{}\omega\in\Omega:E_{P}\big{[}\frac{dS}{dP}|\mathcal{F}_{t}\big{]}(\omega)>0\big{\\}}$.

We recall from Proposition 2.13 and Corollary 2.14 of Bion-Nadal and Di Nunno
[7] that, for fully-dynamic risk measures, strong time-consistency implies
order time-consistency; furthermore, for normalized fully-dynamic risk
measures strong time-consistency is equivalent to the restriction property
plus order time-consistency.

It is worth emphasizing that, differently from strong time-consistency, the
formulation of weak time-consistency is in terms of risk measures with the
same time horizon $u$. The notion of weak time-consistency is motivated by the
discussion done by Bion-Nadal and Di Nunno [7], where it was shown that risk
indifference pricing does not satisfy strong time-consistency, but only a
weaker version.

Notice that under the additional assumption of normalization (3), weak time-
consistency (10) becomes

$\rho_{su}(-\rho_{tu}(X))=\rho_{su}(X)\quad\mbox{ for any }X\in
L^{p}(\mathcal{F}_{u}),$ (13)

while, under both normalization and the restriction property, weak time-
consistency (10) reduces to the classical strong time-consistency (8).

Again from [7] and from the arguments above, it emerges that the restriction
property has not only an extremely important role on time-consistency but also
on risk measures. Indeed, once the restriction property is dropped, we can
explicitly quantify the increasing cost of riskiness for longer time horizons.

###### Definition 3

A fully-dynamic risk measures $(\rho_{st})_{s,t}=(\rho_{st})_{0\leq s\leq
t\leq T}$ is said to take into account _Horizon longevity_ (or _h-longevity_ ,
for short) if for any fixed $s\in[0,T]$,

$\rho_{st}(X)\leq\rho_{su}(X)\quad\mbox{ for any }0\leq t\leq u,\;X\in
L^{p}(\mathcal{F}_{t}).$ (14)

Naturally, a risk measure satisfying restriction property (see (2)) trivially
satisfies h-longevity. Then we talk about _strict h-longevity_ to exclude this
trivial case.

Differently from weak time-consistency (10) that has an impact on the first
time parameter, horizon longevity (14) focuses on the behavior of the second
time parameter, that is the time horizon. Quantifying horizon risk is
particularly important in life insurance and pension funds, as reported in the
introduction of this work.

Our aim is to investigate weak time-consistency and h-longevity on fully-
dynamic risk measures $(\rho_{st})_{0\leq s\leq t\leq T}$ from an axiomatic
point of view as well as their impact on the relation between fully-dynamic
risk measures and BSDEs or BSVIEs.

###  2.1 Time-consistency

The next results focus on weak time-consistency and on its characterizations.
Also we introduce the new concept of sub time-consistency, see Proposition 5,
which will turn out to interplay with h-longevity.

###### Proposition 4

A fully-dynamic risk measure $(\rho_{st})_{0\leq s\leq t\leq T}$ is weakly
time-consistent if and only if it satisfies order time-consistency.

Proof. Assume order time-consistency. Let $X\in L^{p}(\mathcal{F}_{u})$ and
$t\leq u$ be arbitrary. By $\mathcal{F}_{t}$-translation invariance of
$\rho_{tu}$ it follows

$\rho_{tu}(X)=\rho_{tu}(0-\rho_{tu}(X))-\rho_{tu}(0)=\rho_{tu}(\rho_{tu}(0)-\rho_{tu}(X)).$
(15)

Since $Y\triangleq\rho_{tu}(0)-\rho_{tu}(X)\in L^{p}(\mathcal{F}_{t})$, (15)
and order time-consistency imply that $\rho_{su}(X)=\rho_{su}(Y)$ for any
$s\leq t$. The thesis is therefore proved.

Conversely, suppose that $\rho_{tu}(X)=\rho_{tu}(Y)$ holds for some $X,Y\in
L^{p}(\mathcal{F}_{u})$ and $t\leq u$. By weak time-consistency it follows
that

$\rho_{su}(X)=\rho_{su}(\rho_{tu}(0)-\rho_{tu}(X))=\rho_{su}(\rho_{tu}(0)-\rho_{tu}(Y))=\rho_{su}(Y)$

for any $s\leq t$.

Naturally, since strong time-consistency implies order time-consistency, the
result above gives that strong time-consistency implies also weak time-
consistency.

###### Proposition 5

Let $(\alpha_{st})_{0\leq s\leq t\leq T}$ be the minimal penalty terms of
$(\rho_{st})_{0\leq s\leq t\leq T}$.

a) If $(\alpha_{ru})_{0\leq r\leq u\leq T}$ satisfies, for all $s\leq t\leq
u$,

$\alpha_{su}(S)=\alpha_{su}(Q)+E_{Q}[\alpha_{tu}(R)-\operatorname{ess.\\!inf}_{\bar{R}\in\mathcal{Q}_{tu}}\alpha_{tu}(\bar{R})|\mathcal{F}_{s}],$
(16)

for all $Q\in\mathcal{Q}_{su},R\in{Q}_{tu},$ where
$S=Q|_{\mathcal{F}_{t}}\cdot R\in\mathcal{Q}_{su}$ is obtained by pasting $Q$
on $[s,t]$ and $R$ on $[t,u]$, then $(\rho_{st})_{0\leq s\leq t\leq T}$ is
weakly time-consistent.

b) Weak time-consistency of $(\rho_{st})_{0\leq s\leq t\leq T}$ implies that,
for all $s\leq t\leq u$,

$\alpha_{su}(S)\leq\alpha_{su}(Q)+E_{Q}[\alpha_{tu}(R)-\operatorname{ess.\\!inf}_{\bar{R}\in\mathcal{Q}_{tu}}\alpha_{tu}(\bar{R})|\mathcal{F}_{s}],$
(17)

for all $Q\in\mathcal{Q}_{su},R\in{Q}_{tu},$ where
$S=Q|_{\mathcal{F}_{t}}\cdot R\in\mathcal{Q}_{su}$ is obtained by pasting $Q$
on $[s,t]$ and $R$ on $[t,u]$. In (17) equality holds at least for the optimal
scenarios in the dual representation (4)-(5).

It is worth emphasizing that (16) is different from the usual cocycle
condition (11) for two reasons. First, it depends on the additional term
$\operatorname{ess.\\!inf}_{\bar{R}\in\mathcal{Q}_{tu}}\alpha_{tu}(\bar{R})$,
which is due to the non-normalization of the risk measure. Second, both
$\alpha_{su}(S)$ and $\alpha_{su}(Q)$ refer to the time horizon $u$. Note that
$\operatorname{ess.\\!inf}_{\bar{R}\in\mathcal{Q}_{tu}}\alpha_{tu}(\bar{R})\in
L^{p}(\mathcal{F}_{t})$ because of
$\rho_{tu}(0)=-\operatorname{ess.\\!inf}_{\bar{R}\in\mathcal{Q}_{tu}}\alpha_{tu}(\bar{R})$
and, by assumption, $\rho_{tu}(0)\in L^{p}(\mathcal{F}_{t})$.

In terms of the normalized risk measure

$\widetilde{\rho}_{st}(X)\triangleq\rho_{st}(X)-\rho_{st}(0)$

and of its minimal penalty function

$\widetilde{\alpha}_{st}(Q)\triangleq\alpha_{st}(Q)-\operatorname{ess.\\!inf}_{\bar{R}\in\mathcal{Q}_{st}}\alpha_{st}(\bar{R}),$

condition (16) becomes

$\widetilde{\alpha}_{su}(S)=\widetilde{\alpha}_{su}(Q)+E_{Q}[\widetilde{\alpha}_{tu}(R)|\mathcal{F}_{s}],\quad\forall
s,t,u,Q\in\mathcal{Q}_{su},R\in{Q}_{tu}.$ (18)

Notice also that while weak time-consistency of a fully-dynamic risk measure
$(\rho_{st})_{0\leq s\leq t\leq T}$ is equivalent to that of the corresponding
normalized $(\widetilde{\rho}_{st})_{0\leq s\leq t\leq T}$, the same is no
more true for h-longevity. The reason is that the normalization terms
$\rho_{st}(0)$ and $\rho_{su}(0)$ are different in general. As pointed out by
Bion-Nadal and Di Nunno [7], Remark 2.12, also strong time-consistency of a
fully-dynamic risk measure is not transferred to the normalized version.

Proof of Proposition 5. In view of the above comments, for simplicity, we
prove the result in terms of the normalized $\widetilde{\rho}$.

a) Assume that (18) holds true. By the dual representation (4) of
$\widetilde{\rho}$,

$\displaystyle\widetilde{\rho}_{su}(-\widetilde{\rho}_{tu}(X))$ (19)
$\displaystyle=$
$\displaystyle\underset{Q\in\mathcal{Q}_{su}}{\operatorname{ess.\\!max}}\;\\{E_{Q}[\widetilde{\rho}_{tu}(X)|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(Q)\\}$
$\displaystyle=$
$\displaystyle\underset{Q\in\mathcal{Q}_{su}}{\operatorname{ess.\\!max}}\;\\{E_{Q}[\underset{R\in\mathcal{Q}_{tu}}{\operatorname{ess.\\!max}}\;\\{E_{R}[-X|\mathcal{F}_{t}]-\widetilde{\alpha}_{tu}(R)\\}|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(Q)\\}$
$\displaystyle=$
$\displaystyle\underset{Q\in\mathcal{Q}_{su},R\in\mathcal{Q}_{tu}}{\operatorname{ess.\\!max}}\;\\{E_{Q}[E_{R}[-X|\mathcal{F}_{t}]|\mathcal{F}_{s}]-E_{Q}[\widetilde{\alpha}_{tu}(R)|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(Q)\\}$
$\displaystyle=$
$\displaystyle\underset{\begin{subarray}{c}S=Q|_{\mathcal{F}_{t}}\cdot R:\\\
Q\in\mathcal{Q}_{su},R\in\mathcal{Q}_{tu}\end{subarray}}{\operatorname{ess.\\!max}}\;\\{E_{S}[-X|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(S)\\}=\widetilde{\rho}_{su}(X),$
(20)

where $S$ is obtained by pasting $Q$ on $[s,t]$ and $R$ on $[t,u]$. Here
above, (19) follows from the same arguments as in [6] and [11], while the
first equality in (20) is due to (18) and to m-stability (12).

b) Assume that weak time-consistency (13) holds for
$(\widetilde{\rho}_{st})_{0\leq s\leq t\leq T}$. On the one hand, again by the
dual representation (4), we have

$\displaystyle\widetilde{\rho}_{su}(-\widetilde{\rho}_{tu}(X))$ (21)
$\displaystyle=$
$\displaystyle\underset{Q\in\mathcal{Q}_{su},R\in\mathcal{Q}_{tu}}{\operatorname{ess.\\!max}}\;\\{E_{Q}[E_{R}[-X|\mathcal{F}_{t}]|\mathcal{F}_{s}]-E_{Q}[\widetilde{\alpha}_{tu}(R)|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(Q)\\}.$

On the other hand, by weak time-consistency,

$\displaystyle\widetilde{\rho}_{su}(-\widetilde{\rho}_{tu}(X))=\widetilde{\rho}_{su}(X)=\underset{S\in\mathcal{Q}_{su}}{\operatorname{ess.\\!max}}\;\\{E_{S}[-X|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(S)\\}$
(22) $\displaystyle=$
$\displaystyle\underset{\begin{subarray}{c}S=Q|_{\mathcal{F}_{t}}\cdot R:\\\
Q\in\mathcal{Q}_{su},R\in\mathcal{Q}_{tu}\end{subarray}}{\operatorname{ess.\\!max}}\;\\{E_{Q}[E_{R}[-X|\mathcal{F}_{t}]|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(S)\\},$

which is due to the fact that, for any $S\in\mathcal{Q}_{su}$, there exist
$Q\in\mathcal{Q}_{su},R\in\mathcal{Q}_{tu}$ such that
$S=Q|_{\mathcal{F}_{t}}\cdot R$ and, vice versa, given
$Q\in\mathcal{Q}_{su},R\in\mathcal{Q}_{tu}$, the pasting
$S=Q|_{\mathcal{F}_{t}}\cdot R\in\mathcal{Q}_{su}$. Since
$\widetilde{\alpha}_{su}(S)$ is the minimal penalty function on $[s,u]$, (21)
and (22) imply that

$\widetilde{\alpha}_{su}(S)\leq\widetilde{\alpha}_{su}(Q)+E_{Q}[\widetilde{\alpha}_{tu}(R)|\mathcal{F}_{s}]$
(23)

for any $Q\in\mathcal{Q}_{su},R\in\mathcal{Q}_{tu}$ and the pasting
$S=Q|_{\mathcal{F}_{t}}\cdot R$.

It remains to prove that (17) holds with an equality at least for the optimal
scenarios in the dual representation. By the arguments above, it is then
enough to prove the reverse inequality in (23). Since in the dual
representation (4) the $\operatorname{ess.\\!max}$ is attained, it follows
that
$\widetilde{\rho}_{su}(-\widetilde{\rho}_{tu}(X))=\widetilde{\rho}_{su}(X)$
becomes

$E_{Q}[E_{R}[-X|\mathcal{F}_{t}]-\widetilde{\alpha}_{tu}(R)|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(Q)=\widetilde{\rho}_{su}(X)$
(24)

for some $R\in\mathcal{Q}_{tu},Q\in\mathcal{Q}_{su}$ (depending on $X$). By
defining $S$ as the pasting $Q\cdot R$ of $Q$ on $[s,t]$ and $R$ on $[t,u]$,
(24) reduces to

$E_{S}[-X|\mathcal{F}_{s}]-E_{Q}[\widetilde{\alpha}_{tu}(R)|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(Q)=\widetilde{\rho}_{su}(X).$
(25)

From $\widetilde{\rho}_{su}(X)\geq
E_{S}[-X|\mathcal{F}_{s}]-\widetilde{\alpha}_{su}(S)$ and (25), it follows
that
$\widetilde{\alpha}_{su}(S)\geq\widetilde{\alpha}_{su}(Q)+E_{Q}[\widetilde{\alpha}_{tu}(R)|\mathcal{F}_{s}]$.

###### Remark 6

If $(\rho_{st})_{0\leq s\leq t\leq T}$ satisfies weak time-consistency,
h-longevity, and $\rho_{tu}(0)\leq 0$ for any $t,u$, then

$\rho_{st}(-\rho_{tu}(X))\leq\rho_{su}(X)\quad\mbox{ for any }X\in
L^{p}(\mathcal{F}_{u}),$ (26)

called sub (strong) time-consistency in the following. Notice that
$\rho_{tu}(0)\leq 0$ is equivalent to
$\operatorname{ess.\\!inf}_{Q\in\mathcal{Q}_{tu}}\alpha_{tu}(Q)\geq 0$.

On the other hand, sub (strong) time-consistency (26) together with
$\rho_{tu}(0)\geq 0$, for any $t,u$, imply h-longevity. For any $\bar{X}\in
L^{p}(\mathcal{F}_{t})$, indeed,

$\rho_{st}(\bar{X})\leq\rho_{st}(\bar{X}-\rho_{tu}(0))=\rho_{st}(-\rho_{tu}(\bar{X}))\leq\rho_{su}(\bar{X}),$

where the first inequality is due to $\rho_{tu}(0)\geq 0$ and monotonicity,
the last to sub time-consistency, while the equality follows by
$\mathcal{F}_{t}$-translation invariance.

Proceeding similarly to Acciaio and Penner [1], it is easy to prove the
relation between sub time-consistency and acceptance sets for fully-dynamic
risk measures, here formulated as follows.

###### Proposition 7

For any fully-dynamic risk measure $(\rho_{st})_{0\leq s\leq t\leq T}$ the
following are equivalent:

(i) Sub (strong) time-consistency, that is
$\rho_{st}(-\rho_{tu}(X))\leq\rho_{su}(X)$ for any $X\in
L^{p}(\mathcal{F}_{u})$, $s\leq t\leq u$;

(ii) $\mathcal{A}_{su}\subseteq\mathcal{A}_{st}+\mathcal{A}_{tu}$ where
$\mathcal{A}_{su}\triangleq\\{Z\in L^{p}(\mathcal{F}_{u}):\rho_{su}(Z)\leq
0,P\mbox{-a.s.}\\}$ is the acceptance set associated to $\rho_{su}$;

(iii)
$\alpha_{su}(Q)\leq\alpha_{st}(Q|_{\mathcal{F}_{t}})+E_{Q}[\alpha_{tu}(Q)|\mathcal{F}_{s}]$
for any $Q\in\mathcal{Q}_{su}$, $s\leq t\leq u$.

To summarize, the following implications among the different notions of time-
consistency investigated above and h-longevity hold:

$\begin{array}[]{cccl}&&\mbox{\small strong TC}&\\\ \\\
&&\Updownarrow&{\footnotesize\mbox{+ normalization and restriction}}\\\
&&&{\footnotesize\mbox{\quad(see~{}\cite[cite]{[\@@bibref{}{bion-nadal-di-
nunno}{}{}]})}}\\\ \\\ \mbox{\small order TC}&\Longleftrightarrow&\mbox{\small
weakly TC}&\\\ &{\footnotesize\mbox{\quad(see Prop.~{}\ref{prop: equiv-weak-
tc})}}&&\\\ &&\Downarrow&{\footnotesize\mbox{+ h-longevity and
}\rho_{tu}(0)\leq 0\mbox{ }}\\\ &&&{\footnotesize\mbox{\quad(see
Remark~{}\ref{rem: tc and longevity})}}\\\ &&&\\\ &&\mbox{\small sub TC}&\\\
\\\ &&\Downarrow&{\footnotesize\mbox{+ }\rho_{tu}(0)\geq 0}\\\
&&&\mbox{\footnotesize\quad(see Remark~{}\ref{rem: tc and longevity})}\\\
&&\mbox{\small h-longevity}&\end{array}$

###  2.2 Horizon risk and h-longevity

We recall that h-longevity (14) has been formulated as

$\rho_{st}(X)\leq\rho_{su}(X)\quad\mbox{ for any }0\leq s\leq t\leq u,X\in
L^{p}(\mathcal{F}_{t}).$

From the formulation of h-longevity it is clear that the risk measure with
longer horizon $\rho_{su}$ is relevant only restricted on
$L^{p}(\mathcal{F}_{t})$. In other words, longevity aims to penalize the risk
measurement of a position done with the wrong risk measure, i.e. with the risk
measure suitable for a longer horizon.

For this reason, in the sequel, we denote the restriction of $\rho_{su}$ on
$L^{p}(\mathcal{F}_{t})$ by $\bar{\rho}^{t}_{su}$, while
$\bar{\alpha}^{t}_{su}$ is the corresponding minimal penalty function, i.e.

$\bar{\alpha}^{t}_{su}(Q)=\underset{X\in
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\;\\{E_{Q}[-X|\mathcal{F}_{s}]-\bar{\rho}^{t}_{su}(X)\\},\quad
Q\in\mathcal{Q}_{st}.$

It then follows that

$\bar{\alpha}^{t}_{su}(Q_{|\mathcal{F}_{t}})\leq\alpha_{su}(Q),\qquad
Q\in\mathcal{Q}_{su}.$

In fact,

$\displaystyle\alpha_{su}(Q)$ $\displaystyle=$ $\displaystyle\underset{X\in
L^{p}(\mathcal{F}_{u})}{\operatorname{ess.\\!sup}}\\{E_{Q}[-|X\mathcal{F}_{s}]-\rho_{su}(X)\\}$
$\displaystyle\geq$ $\displaystyle\underset{X\in
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\\{E_{Q}[-X|\mathcal{F}_{s}]-\bar{\rho}^{t}_{su}(X)\\}=\bar{\alpha}^{t}_{su}(\left.Q\right|_{\mathcal{F}_{t}}).$

Using the notation above, h-longevity naturally leads to

$\bar{\rho}^{t}_{su}(X)=\rho_{su}(X)=\rho_{st}(X)+\gamma(s,t,u,X)$ (27)

for any $0\leq s\leq t\leq u$, $X\in L^{p}(\mathcal{F}_{t})$, and for a
suitable $\mathcal{F}_{s}$-measurable $\gamma(s,t,u,X)\geq 0$ that may depend
on the position $X$, on the time of evaluation $s$, on the time parameter $t$
referring to the measurability of $X$, and on the “wrong” time horizon $u$
used for evaluating $X$. The term $\gamma$ can be then seen as an indicator
quantifying the horizon risk or, roughly speaking, as an additive term of
adjustment/calibration. It follows that, for any $s\leq t\leq u$ and $X\in
L^{p}(\mathcal{F}_{t})$, $\gamma(s,t,t,X)=0$ and, by translation invariance of
the fully-dynamic risk measure, that $\gamma(s,t,u,X+c_{s})=\gamma(s,t,u,X)$
for any $c_{s}\in L^{p}(\mathcal{F}_{s})$. The restriction property (2) is
then equivalent to $\gamma(s,t,u,X)=0$ for all $0\leq s\leq t\leq u$, $X\in
L^{p}(\mathcal{F}_{t})$.

The following result characterizes longevity in terms of acceptance sets and
of penalty functions.

###### Proposition 8

For a fully dynamic risk measure $(\rho_{st})_{s,t}$:

(a) longevity is equivalent to $\mathcal{A}_{su}\cap
L^{p}(\mathcal{F}_{t})\subseteq\mathcal{A}_{st}$ for any $s\leq t\leq u$.

(b) restriction is equivalent to $\mathcal{A}_{su}\cap
L^{p}(\mathcal{F}_{t})=\mathcal{A}_{st}$ for any $s\leq t\leq u$.

(c) longevity implies that $\alpha_{st}(Q)\geq\bar{\alpha}^{t}_{su}(Q)$ and
also

$\bar{\alpha}^{t}_{su}(Q)\geq\alpha_{st}(Q)-\operatorname{ess.\\!sup}_{X\in
L^{p}(\mathcal{F}_{t})}\gamma(s,t,u,X),$

for any $s\leq t\leq u$ and $Q\in\mathcal{Q}_{st}$.

(d) restriction implies that $\alpha_{st}(Q)=\bar{\alpha}^{t}_{su}(Q)$ for any
$s\leq t\leq u$ and $Q\in\mathcal{Q}_{st}$.

Proof. (a) For any $X\in\mathcal{A}_{su}\cap L^{p}(\mathcal{F}_{t})$ it
follows that $\rho_{su}(X)\leq 0$ and, by longevity, also that
$\rho_{st}(X)\leq 0$. Hence $\mathcal{A}_{su}\cap
L^{p}(\mathcal{F}_{t})\subseteq\mathcal{A}_{st}$. Conversely, from the
representation of risk measures in terms of acceptance sets and from the
condition on acceptance sets, it follows that, for any $X\in
L^{p}(\mathcal{F}_{t})$,

$\displaystyle\rho_{su}(X)$ $\displaystyle=$
$\displaystyle\operatorname{ess.\\!inf}\\{m_{s}\in
L^{p}(\mathcal{F}_{s})|\;m_{s}+X\in\mathcal{A}_{su}\\}$ $\displaystyle\geq$
$\displaystyle\operatorname{ess.\\!inf}\\{m_{s}\in
L^{p}(\mathcal{F}_{s})|\;m_{s}+X\in\mathcal{A}_{st}\\}=\rho_{st}(X).$

(b) Assume now restriction. From (27), we see that restriction can be
interpreted as h-longevity with $\gamma(s,t,u,X)=0$. Hence, from (a), it
remains to prove the reverse inclusion for acceptance sets. For any
$X\in\mathcal{A}_{st}$ (hence $X\in L^{p}(\mathcal{F}_{t})$) we have that
$\rho_{st}(X)\leq 0$. By restriction, $\rho_{su}(X)=\rho_{st}(X)\leq 0$, hence
$\mathcal{A}_{su}\cap L^{p}(\mathcal{F}_{t})\supseteq\mathcal{A}_{st}$.
Conversely, assume $\mathcal{A}_{su}\cap
L^{p}(\mathcal{F}_{t})=\mathcal{A}_{st}$. From the representation of risk
measures in terms of acceptance sets, we obtain that, for any $X\in
L^{p}(\mathcal{F}_{t})$,

$\displaystyle\rho_{su}(X)$ $\displaystyle=$
$\displaystyle\operatorname{ess.\\!inf}\\{m_{s}\in
L^{p}(\mathcal{F}_{s})|\>m_{s}+X\in\mathcal{A}_{su}\\}$ $\displaystyle=$
$\displaystyle\operatorname{ess.\\!inf}\\{m_{s}\in
L^{p}(\mathcal{F}_{s})|\>m_{s}+X\in\mathcal{A}_{st}\\}=\rho_{st}(X).$

(c) Proceeding similarly to Föllmer and Schied [15], Thm. 4.16, by translation
invariance of $\rho_{su}$, we have

$\alpha_{st}(Q)=\underset{X\in\mathcal{A}_{st}}{\operatorname{ess.\\!sup}}\,E_{Q}[-X|\mathcal{F}_{s}]$

for any $s\leq t$ and $Q\in\mathcal{Q}_{st}$. By (a), h-longevity and
translation invariance imply that, for any $Q\in\mathcal{Q}_{st}$,

$\displaystyle\alpha_{st}(Q)$ $\displaystyle=$
$\displaystyle\underset{X\in\mathcal{A}_{st}}{\operatorname{ess.\\!sup}}\,E_{Q}[-X|\mathcal{F}_{s}]$
$\displaystyle\geq$ $\displaystyle\underset{X\in\mathcal{A}_{su}\cap
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}E_{Q}[-X|\mathcal{F}_{s}]$
$\displaystyle=$ $\displaystyle\underset{\begin{subarray}{c}X\in
L^{p}(\mathcal{F}_{t}):\\\ \rho_{su}(X)\leq
0\end{subarray}}{\operatorname{ess.\\!sup}}E_{Q}[-X|\mathcal{F}_{s}]=\bar{\alpha}^{t}_{su}(Q).$

Moreover, in terms of the indicator $\gamma$ in (27), we have that

$\displaystyle\bar{\alpha}^{t}_{su}(Q)$ $\displaystyle=$
$\displaystyle\underset{X\in
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\\{E_{Q}[-X|\mathcal{F}_{s}]-\bar{\rho}^{t}_{su}(X)\\}$
$\displaystyle=$ $\displaystyle\underset{X\in
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\\{E_{Q}[-X|\mathcal{F}_{s}]-\rho_{st}(X)-\gamma(s,t,u,X)\\}$
$\displaystyle\geq$ $\displaystyle\underset{X\in
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\\{E_{Q}[-X|\mathcal{F}_{s}]-\rho_{st}(X)\\}-\underset{X\in
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\gamma(s,t,u,X)$
$\displaystyle=$ $\displaystyle\alpha_{st}(Q)-\underset{X\in
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\gamma(s,t,u,X).$

(d) By the equivalence between restriction and the condition on acceptance
sets in (b), it follows that for any $Q\in\mathcal{Q}_{st}$

$\displaystyle\alpha_{st}(Q)$ $\displaystyle=$
$\displaystyle\underset{X\in\mathcal{A}_{st}}{\operatorname{ess.\\!sup}}\,E_{Q}[-X|\mathcal{F}_{s}]$
$\displaystyle=$ $\displaystyle\underset{X\in\mathcal{A}_{su}\cap
L^{p}(\mathcal{F}_{t})}{\operatorname{ess.\\!sup}}\,E_{Q}[-X|\mathcal{F}_{s}]=\bar{\alpha}^{t}_{su}(Q).$

The proof is then complete.

With (c), we see that the greater is $\operatorname{ess.\\!sup}_{X\in
L^{p}(\mathcal{F}_{t})}\gamma(s,t,u,X)$ for $\mathcal{F}_{t}$-measurable
positions, the lower is the lower bound
$\alpha_{st}(\left.Q\right|_{\mathcal{F}_{t}})-\operatorname{ess.\\!sup}_{X\in
L^{p}(\mathcal{F}_{t})}\gamma(s,t,u,X)$ of $\alpha_{su}(Q)$. The term
$\operatorname{ess.\\!sup}_{X\in L^{p}(\mathcal{F}_{t})}\gamma(s,t,u,X)$ can
be then interpreted as the maximal error (at the level of the penalty
function) for a wrong use of $\rho_{su}$ for positions $X$ belonging to
$L^{p}(\mathcal{F}_{t})$, that is, $\rho_{s\cdot}$ with a wrong time horizon.

From a motivational point of view, one can imagine $\gamma$ dependent on the
time horizon, that is, $\gamma(s,t,u,X)=\gamma_{s,t}(h,X)$ with $h=u-t$ or,
even, $\gamma_{s,t}(h)$ independent from $X$. In particular, $h=u-t$ can be
interpreted as the length of the time interval over which there is an
uncorrect use of the risk measure ($\rho_{su}$ versus $\rho_{st}$). See
Example 12 and Example 26.

In what follows, we investigate whether BSDEs or BSVIEs can provide families
of fully-dynamic risk measures satisfying sub or weak time-consistency and
h-longevity. In particular, we study the cases of a single driver $g$ and of a
family of drivers $(g_{t})_{t\in[0,T]}$ depending on the time horizon $t$
considered in $\rho_{st}$. The former case will be shortened by BSDE $(g)$ (or
BSVIE $(g)$), the latter by BSDE $(g_{t})$ (or BSVIE $(g_{t})$).

##  3 Relation with BSDEs

Hereafter, we consider a fully-dynamic risk measure induced by a single BSDE
with driver $g$ and also measures generated by a family of BSDEs associated to
the drivers $\mathcal{G}=(g_{t})_{t\in[0,T]}$ where the index $t$ refers to
the time horizon. We will restrict our attention to $L^{2}$ spaces.

On the probability space $(\Omega,\mathcal{F},P)$ we consider a
$d$-dimensional Brownian motion $(B_{t})_{t\in[0,T]}$ and the $P$-augmented
natural filtration $(\mathcal{F}_{t})_{t\in[0,T]}$ of $(B_{t})_{t\in[0,T]}$.
According to Peng [21], the solution $(Y_{t},Z_{t})_{t\in[0,T]}$ of the BSDE

$Y_{t}=X+\int_{t}^{T}g(s,Y_{s},Z_{s})\,ds-\int_{t}^{T}Z_{s}\,dB_{s}$ (28)

can be seen as an operator depending on the driver $g$ and evaluated at the
final condition $X\in L^{2}(\mathcal{F}_{T})$. In our work, we consider a
driver

$g:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{R}$

satisfying the standard assumptions:

  * •

adapted,

  * •

uniformly Lipschitz, i.e. there exists a constant $C>0$ such that, $dP\times
dt$-a.e.,

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

for any $y_{1},y_{2}\in\mathbb{R},\,z_{1},z_{2}\in\mathbb{R}^{d}$, where
$|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^{k}$ for whatever $k$ is
relevant;

  * •

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

Under these conditions, equation (28) admits a unique solution
$(Y_{t},Z_{t})_{t\in[0,T]}$, with
$(Y_{t})_{t\in[0,T]}\in\mathbb{H}^{2}_{[0,T]}(\mathbb{R})$ and
$(Z_{t})_{t\in[0,T]}\in\mathbb{H}^{2}_{[0,T]}(\mathbb{R}^{d})$, where we have
set

$\mathbb{H}^{2}_{[a,b]}(\mathbb{R}^{k})\triangleq\Big{\\{}\mbox{adapted
}\mathbb{R}^{k}\mbox{-valued processes
}(\eta_{s})_{s\in[a,b]}:E\Big{[}\int_{a}^{b}|\eta_{s}|^{2}\,ds\Big{]}<\infty\Big{\\}}.$

For further details see, e.g., El Karoui et al. [13].

In Peng [21, 22] and Rosazza Gianin [25] the relationship among BSDEs,
nonlinear expectations and dynamic risk measures is detailed clarifying that
the properties of $g$ reflect the properties of the risk measures associated.
To summarize, in the financial context of monetary risk measures, if $g$ in
(28) does not depend on $y$, then the risk measure is translation invariant.
If $g$ depends monotonically on $y$ then the risk measure is cash-subadditive,
as discussed in El Karoui and Ravanelli [14]. Also $g$ convex produces a
convex risk measure. See Barrieu and El Karoui [3], Jiang [18], and Rosazza
Gianin [25].

Beyond the Lipschitz condition on the driver, the connection between BSDEs and
risk measures has been studied in terms of maximal solutions of BSDEs (see,
e.g., Barrieu and El Karoui [3] and Kobylanski [19]). When it comes to
generalizations beyond the Brownian framework we can refer, e.g., to Royer
[26], Quenez and Sulem [23], and Laeven and Stadje [20].

###  3.1 Risk measures generated by a single BSDE

In this work we have assumed cash-invariant risk measures, then we consider
the following BSDE with driver $g$, not depending on $y$, and terminal
condition $X\in L^{2}(\mathcal{F}_{u})$:

$Y_{t}=X+\int_{t}^{u}g(s,Z_{s})ds-\int_{t}^{u}Z_{s}dB_{s},\quad 0\leq t\leq
u,$ (29)

where $u\in[0,T]$. To simplify the notation, we will occasionally adopt the
Peng’s notation where $\mathcal{E}^{g}\left(X|\mathcal{F}_{t}\right)$ denotes
the conditional $g$-expectation of $X$ at time $t$, that is the $Y$-component
of the solution $(Y,Z)$ at time $t$ of the BSDE above.Then, we focus on risk
measures of the following form:

$\rho_{tu}(X)=\mathcal{E}^{g}\left(-X|\mathcal{F}_{t}\right),\quad X\in
L^{2}(\mathcal{F}_{u}).$ (30)

The condition $g(t,0)=0$, guarantees the restriction property as well as the
normalization (3). Hence, in our case we cannot assume $g(t,0)=0$.

###### Proposition 9

The following statements are equivalent:

(a) $g(t,0)=0$ for any $t\in[0,T]$;

(b) $\rho$ is normalized, see (3);

(c) $\rho$ has the restriction property (2).

Proof. (a) $\Rightarrow$ (b). This implication follows immediately by
remarking that $(Y_{t},Z_{t})$ with $Y_{t}=Z_{t}=0$ for any $t$ is the unique
solution when the terminal condition is $0$. See also Peng [21].

(b) $\Rightarrow$ (a). Assume that $\rho_{tu}(0)=0$ for any $0\leq t\leq u\leq
T$. It follows that

$0=Y_{t}=\int_{t}^{u}g(s,Z_{s})ds-\int_{t}^{u}Z_{s}dB_{s}$

holds for any $0\leq t\leq u\leq T$. Hence

$\int_{t}^{u}g(s,Z_{s})ds=\int_{t}^{u}Z_{s}dB_{s},\quad\mbox{ for any }0\leq
t\leq u\leq T.$

Since a continuous martingale and a process of finite variation can be equal
only if the martingale is constant (see Prop. 1.2 in Chapter 4 of Revuz and
Yor [24]), then for any $t$ the martingale $M_{t}(u)=\int_{t}^{u}Z_{s}dB_{s}$,
$u\geq t$, starting from $0$ is identically equal to $0$. Hence $Z_{s}\equiv
0$ for any $s\geq t$ and $\int_{t}^{u}g(s,Z_{s})ds\equiv 0$ for any $u\geq t$.
By replacing $Z_{s}$ and deriving this last equation with respect to $u$, it
follows that $g(u,0)=0$ for any $u$.

(a) $\Rightarrow$ (c). This implication was proved in Peng [21]. Assume that
$X\in L^{2}(\mathcal{F}_{u})$ and consider $\rho_{tu}(X)$ and $\rho_{tv}(X)$
for any $v\geq u$. Let us denote by $(Y^{X,u}_{r},Z^{X,u}_{r})$ (resp.
$(Y^{X,v}_{r},Z^{X,v}_{r})$) the solution corresponding to $\rho_{tu}(X)$
(resp. $\rho_{tv}(X)$) at time $r\leq u$. Since $(Y^{X,v}_{r},Z^{X,v}_{r})$
with

$Y^{X,v}_{r}=\left\\{\begin{array}[]{rl}Y^{X,u}_{r};&r\leq u\\\ -X;&u<r\leq
v\end{array}\right.\quad
Z^{X,v}(r,s)=\left\\{\begin{array}[]{rl}Z^{X,u}_{r};&s\leq u\\\ 0;&u<s\leq
v\end{array}\right.$

is a solution of $\rho_{tv}(X)$ when $g(t,0)=0$ for any $t\in[0,T]$, the
restriction property follows.

(c) $\Rightarrow$ (a). By translation invariance and restriction, it holds
that for any $X\in L^{2}(\mathcal{F}_{t})\subseteq L^{2}(\mathcal{F}_{u})$

$-X+\rho_{tT}(0)=\rho_{tT}(X)=\rho_{tu}(X)=-X+\rho_{tu}(0).$

Hence $\rho_{tT}(0)=\rho_{tu}(0)$ for any $t\leq u\leq T$. Since $\rho_{tu}$
is induced by the BSDE (29) via (30), it follows that $\rho_{tt}(0)=0$.
Consequently, $\rho_{tu}(0)=0$ for any $t\leq u\leq T$. By this the proof is
complete.

Note that while normalization refers to a single risk measure $\rho_{tu}$,
restriction involves the whole family $(\rho_{tu})_{t,u}$. It is already known
that any BSDE satisfies both weak and strong time-consistency (see Barrieu and
El Karoui [3], El Karoui et al. [13], Bion Nadal and Di Nunno [7]). From the
arguments above, any BSDE with a driver $g$ such that $g(t,0)=0$ for any
$t\in[0,T]$ satisfies normalization, restriction and strong time-consistency.
Thus, it also trivially satisfies h-longevity.

Here below, we provide some examples where the driver is not normalized and
$(\rho_{tu})_{t,u}$ does not satisfies restriction, but only (strict)
h-longevity.

###### Example 10

a) Consider the driver $g(t,z)=a$ for any $t\in[0,T],z\in\mathbb{R}^{d}$, with
$a\in\mathbb{R}\setminus\\{0\\}$. It can be checked easily that, for any
$t\in[0,T]$ and $X\in L^{2}(\mathcal{F}_{T})$,

$\rho_{tT}(X)=E_{P}\left[\left.-X+\int_{t}^{T}a\,ds\right|\mathcal{F}_{t}\right]=E_{P}\left[\left.-X\right|\mathcal{F}_{t}\right]+(T-t)a.$

This means that, for $a\neq 0$, $(\rho_{tu})_{t,u}$ is not normalized and does
not satisfy the restriction property. Instead, it satisfies h-longevity
whenever $a>0$.

b) Consider now the following driver: $g(t,z)=bz+a$ for any
$t\in[0,T],z\in\mathbb{R}^{d}$, with $a,b\in\mathbb{R}\setminus\\{0\\}$. By
Girsanov Theorem,

$\displaystyle-dY_{t}$ $\displaystyle=(bZ_{t}+a)dt-Z_{t}\,dB_{t}$
$\displaystyle=a\,dt-Z_{t}\,dB_{t}^{Q}$

where
$E\Big{[}\frac{dQ}{dP}\Big{|}\mathcal{F}_{t}\Big{]}=\exp\left\\{-\frac{1}{2}b^{2}t+b\cdot
B_{t}\right\\}$. It then follows easily that

$\rho_{tT}(X)=E_{Q}\left[\left.-X\right|\mathcal{F}_{t}\right]+(T-t)\,a.$

As before, for $a\neq 0$, $(\rho_{tu})_{t,u}$ is not normalized and does not
satisfy the restriction property. Instead, it satisfies h-longevity whenever
$a=g(t,0)>0$.

The next result provides a sufficient condition on the driver $g$ for
h-longevity of $(\rho_{tu})_{t,u}$.

###### Proposition 11

If $g(v,0)\geq 0$ for any $v\in[0,T]$, then h-longevity holds. Furthermore,
for all $s,u\in[0,T]$, $s\leq u$,
$\gamma(s,t,u,X)=E_{\widetilde{Q}_{X}}\left[\int_{t}^{u}g(v,0)dv|\mathcal{F}_{s}\right]$,
$s\leq t\leq u,X\in L^{p}(\mathcal{F}_{t})$, where $\widetilde{Q}_{X}$ is a
probability measure on $\mathcal{Q}_{su}$ depending on $X$ equivalent to $P$,
with density

$\frac{d\widetilde{Q}_{X}}{dP}=\exp\left\\{-\frac{1}{2}\int_{s}^{u}|\Delta_{z}g(v)|^{2}dv+\int_{s}^{u}\Delta_{z}g(v)dB_{v}\right\\}.$

Here above $\Delta_{z}g(v)=(\Delta_{z}^{i}g(v))_{i=1,...,d}$ and

$\Delta_{z}^{i}g(v)\triangleq\frac{g(v,Z_{v}^{u})-g(v,\bar{Z}_{v}^{t})}{d(Z_{v}^{u,i}-\bar{Z}_{v}^{t,i})}1_{\\{Z_{v}^{u,i}\neq\bar{Z}_{v}^{t,i}\\}}.$

The probability measure ${\widetilde{Q}_{X}}$ here above can be interpreted as
an h-longevity premium measure.

Proof. Let $s\leq t\leq u$ and $X\in L^{p}(\mathcal{F}_{t})$. The risk
measures $\rho_{st}(X)$ and $\rho_{su}(X)$ satisfy the following BSDEs:

$\displaystyle\rho_{st}(X)$ $\displaystyle=$
$\displaystyle-X+\int_{s}^{t}g(v,Z_{v}^{t})dv-\int_{s}^{t}Z_{v}^{t}dB_{v}$
$\displaystyle\rho_{su}(X)$ $\displaystyle=$
$\displaystyle-X+\int_{s}^{u}g(v,Z_{v}^{u})dv-\int_{s}^{u}Z_{v}^{u}dB_{v},$

respectively. Set now the $\mathbb{R}^{d}$-valued process

$\bar{Z}_{v}^{t}=\left\\{\begin{array}[]{rl}Z_{v}^{t};&v\leq t\\\ 0;&t<v\leq
u\end{array}\right.;\quad\widetilde{Z}_{v}=Z_{v}^{u}-\bar{Z}_{v}^{t}.$

Then

$\displaystyle\rho_{su}(X)-\rho_{st}(X)$
$\displaystyle=\int_{s}^{u}[g(v,Z_{v}^{u})-g(v,\bar{Z}_{v}^{t})]dv+\int_{t}^{u}g(v,\bar{Z}_{v}^{t})dv-\int_{s}^{u}[Z_{v}^{u}-\bar{Z}_{v}^{t}]dB_{v}-\int_{t}^{u}\bar{Z}_{v}^{t}dB_{v}$
$\displaystyle=\int_{s}^{u}[g(v,Z_{v}^{u})-g(v,\bar{Z}_{v}^{t})]dv-\int_{s}^{u}\widetilde{Z}_{v}dB_{v}+\int_{t}^{u}g(v,0)dv$
$\displaystyle=\int_{s}^{u}\Delta_{z}g(v)\cdot\widetilde{Z}_{v}dv-\int_{s}^{u}\widetilde{Z}_{v}dB_{v}+\int_{t}^{u}g(v,0)dv.$
(31)

Furthermore, (31) can be rewritten also as

$\delta\rho_{s}=\Gamma^{t,u}+\int_{s}^{u}\Delta_{z}g(v)\cdot\widetilde{Z}_{v}dv-\int_{s}^{u}\widetilde{Z}_{v}dB_{v},$
(32)

where $\delta\rho_{s}\triangleq\rho_{su}(X)-\rho_{st}(X)$ and
$\Gamma^{t,u}\triangleq\int_{t}^{u}g(v,0)dv$ represents the final condition at
time $u$ (which depends on $t$ but not on $s$) of the linear BSDE (32).

Since $\Gamma^{t,u}\geq 0$ for any $t$ by hypothesis and
$\Delta_{z}g(v)\in\mathbb{H}^{2}_{[s,u]}(\mathbb{R}^{d})$ by the assumption of
$g$ Lipschitz in $z$, by Prop. 2.2 of El Karoui, Peng and Quenez [13] it
follows that $\delta\rho_{s}\geq 0$ for any $s\leq t$.

By applying Girsanov Theorem, (31) becomes

$\displaystyle\rho_{su}(X)-\rho_{st}(X)$ $\displaystyle=$
$\displaystyle\int_{s}^{u}\Delta_{z}g(v)\cdot\widetilde{Z}_{v}dv-\int_{s}^{u}\widetilde{Z}_{v}dB_{v}+\int_{t}^{u}g(v,0)dv$
$\displaystyle=$
$\displaystyle-\int_{s}^{u}\widetilde{Z}_{v}dB^{\widetilde{Q}_{X}}_{v}+\int_{t}^{u}g(v,0)dv,$

where $B^{\widetilde{Q}_{X}}_{v}\triangleq
B_{v}-B_{s}-\int_{s}^{v}\Delta_{z}g(r)\,dr$, $v\in[s,u]$, is a
$\widetilde{Q}_{X}$-Brownian motion. Hence, by taking the conditional
expectation with respect to $\widetilde{Q}_{X}$,

$\displaystyle\rho_{su}(X)-\rho_{st}(X)$ $\displaystyle=$ $\displaystyle
E_{\widetilde{Q}_{X}}\left[-\int_{s}^{u}\widetilde{Z}_{v}dB^{\widetilde{Q}_{X}}_{v}+\int_{t}^{u}g(v,0)dv\Big{|}\mathcal{F}_{s}\right]$
$\displaystyle=$ $\displaystyle
E_{\widetilde{Q}_{X}}\left[\int_{t}^{u}g(v,0)dv\Big{|}\mathcal{F}_{s}\right].$

By assumption on $g(\cdot,0)$, it follows that $\rho_{su}(X)-\rho_{st}(X)\geq
0$ and that
$\gamma(s,t,u,X)=\rho_{su}(X)-\rho_{st}(X)=E_{\widetilde{Q}_{X}}\left[\int_{t}^{u}g(v,0)dv|\mathcal{F}_{s}\right]$.

As discussed in Section 2.2, $\gamma$ may depend on the time horizon, that is,
$\gamma(s,t,u,X)=\gamma_{s,t}(h,X)$ with $h=u-t$ or, even, $\gamma_{s,t}(h)$
independent from $X$. The following example provides some cases covering the
situation above.

###### Example 12

Let $g(v,0)\geq 0$ for any $v\in[0,T]$. Hence, by the result above,
h-longevity holds and
$\gamma(s,t,u,X)=E_{\widetilde{Q}_{X}}\left[\int_{t}^{u}g(v,0)dv|\mathcal{F}_{s}\right]$
for any $X\in L^{2}(\mathcal{F}_{t})$.

a) If $g(v,0)=c$ for any $v\in[0,T]$, with $c\geq 0$, then $c$ is necessarily
deterministic (since it should be measurable for any $v\geq 0$) and,
consequently,

$\gamma(s,t,u,X)=E_{\widetilde{Q}_{X}}\left[\int_{t}^{u}g(v,0)dv\Big{|}\mathcal{F}_{s}\right]=(u-t)c.$

Hence $\gamma$ only depends on $h=u-t$, that is, roughly speaking, on the
length of the time interval over which there is an uncorrect use of the risk
measure ($\rho_{su}$ versus $\rho_{st}$).

b) If $g(v,0)=\exp(-r\,v)$ for any $v\in[0,T]$, with $r\geq 0$, then $r$ is
necessarily deterministic (for the same arguments as above) and, consequently,

$\gamma(s,t,u,X)=\frac{e^{-rt}\left(1-e^{-r(u-t)}\right)}{r}$

In other words, $\gamma$ only depends on the “right” time horizon $t$
(referring to the measurability of $X$) and on the length of the time interval
$[t,u]$.

###  3.2 Risk measures generated by a family of BSDEs

Now we consider general risk measures induced by a family of BSDEs of type
(28) with drivers $\mathcal{G}=(g_{u})_{u\in[0,T]}$ depending on the time
horizon $u$ of $\rho_{tu}$. While the focus of this paper is to work with
BSDEs where the driver is independent on $y$ (leading to cash-invariant risk
measures), we still introduce some concepts and results for the general case
which serves the cash-subadditive case of El Karoui and Ravanelli [14].

For later use, by increasing family $\mathcal{G}=(g_{u})_{u\in[0,T]}$ of BSDE
drivers it is meant, for any $t\leq u$,

$g_{t}(v,y,z)\leq g_{u}(v,y,z)\mbox{ for any
}v\in[0,t],y\in\mathbb{R},z\in\mathbb{R}^{d}.$

Suppose that, for any $t\leq u$, the risk measure $\rho_{tu}$ comes from a
$g_{u}$-expectation with a driver depending on the maturity $u$. This means
that

$\rho_{tu}(X)=\rho_{tu}^{\mathcal{G}}(X)=\mathcal{E}^{g_{u}}(-X|\mathcal{F}_{t}),\mbox{
for any }X\in L^{2}(\mathcal{F}_{u}).$ (33)

Assume now that $(g_{u})_{u\in[0,T]}$ is a family of drivers depending on the
maturity $u$, independent of $y$, Lipschitz, and convex in $z$. Then the risk
measure $(\rho^{\mathcal{G}}_{tu})_{t,u}$ defined by (29), (33) satisfies
monotonicity, convexity, continuity from above/below, and translation
invariance. Furthermore, for $u\in[0,T]$, if $g_{u}(v,0)=0$ for any $v\leq u$,
then $\rho^{\mathcal{G}}_{tu}(0)=0$ for any $t\leq u$. In general, however,
$g_{u}(v,0)=0$ for any $v\leq u$ does not imply the restriction property.

In a Brownian setting, $\rho_{tu}^{\mathcal{G}}$ can be represented as

$\rho_{tu}(X)=\rho_{tu}^{\mathcal{G}}(X)=\underset{Q\in\mathcal{Q}_{tu}}{\operatorname{ess.\\!sup}}\,\left\\{E_{Q}\left[-X|\mathcal{F}_{t}\right]-\alpha_{tu}^{\mathcal{G}}(Q)\right\\},\quad
X\in L^{2}(\mathcal{F}_{u}),$

where $\mathcal{Q}_{tu}$ is defined in (6) and the penalty term
$\alpha_{tu}^{\mathcal{G}}$ associated to $\rho_{tu}^{\mathcal{G}}$ is given
by

$\alpha_{tu}^{\mathcal{G}}(Q)=E_{Q}\left[\left.\int_{t}^{u}g^{*}_{u}(v,q_{v})dv\right|\mathcal{F}_{t}\right],$

where $g^{*}_{u}$ denotes the convex conjugate of the function $g_{u}$. See
Delbaen et al. [10].

The following result shows that, for $g_{u}(r,0)=0$ for any $r,u$, the
restriction property is satisfied only for risk measures induced by a BSDE
with a single driver $g$ (that is, constant with respect to the maturity
time).

###### Proposition 13

For any $u\in[0,T]$, let $g_{u}(v,0)=0$ for any $v\leq u$.

The restriction property (2) holds if and only if $g_{u}$ is constant in $u$.

Proof. If $g_{u}$ is constant in $u$, the restriction property follows
directly by Proposition 9.

Conversely, assume that the restriction property holds, i.e.
$\rho_{tu}(X)=\rho_{tv}(X)$ for any $t\leq u\leq v$ and $X\in
L^{2}(\mathcal{F}_{u})$. Proceeding as in the proof of the Converse Comparison
Theorem of Briand et al. [8], Thm. 4.1, and Jiang [18], Lemma 2.1,

$\displaystyle g_{u}(s,z)=\lim_{\varepsilon\to
0}\frac{\rho_{su}(-z\cdot(B_{s+\varepsilon}-B_{\varepsilon}))}{\varepsilon}$
$\displaystyle g_{v}(s,z)=\lim_{\varepsilon\to
0}\frac{\rho_{sv}(-z\cdot(B_{s+\varepsilon}-B_{\varepsilon}))}{\varepsilon}$

with convergence in $L^{p}$ with $p\in[1,2)$, for any $z\in\mathbb{R}^{d}$,
$u\leq v$ and $s\in[0,u]$. By extracting a subsequence to obtain convergence
$P$-a.s. and passing to the limit as $\varepsilon\to 0$, it holds that

$\frac{\rho_{sv}(-z(B_{s+\varepsilon}-B_{\varepsilon}))}{\varepsilon}=\frac{\rho_{su}(-z(B_{s+\varepsilon}-B_{\varepsilon}))}{\varepsilon}\longrightarrow
g_{u}(s,z),\quad\epsilon\to 0,\quad P-a.s.$

where the equality is due to restriction. The thesis then follows because

$\frac{\rho_{sv}(-z(B_{s+\varepsilon}-B_{\varepsilon}))}{\varepsilon}\longrightarrow
g_{v}(s,z),\quad\epsilon\to 0,\quad P-a.s.$

By this we end the proof.

As done in Example 10 for a single driver, we provide here below some examples
where restriction fails.

###### Example 14

a) Consider the driver $g_{u}(t,z)=a_{u}$ for any
$t\in[0,u],z\in\mathbb{R}^{d}$, with $a_{u}\in\mathbb{R}\setminus\\{0\\}$
depending on the maturity $u$. It can be checked easily that, for any
$t\in[0,T]$ and $X\in L^{2}(\mathcal{F}_{T})$,

$\rho_{tu}(X)=E_{P}\left[\left.-X\right|\mathcal{F}_{t}\right]+(u-t)a_{u}.$

This means that, for $a_{u}\neq 0$, $(\rho_{tu})_{t,u}$ is not normalized and
does not satisfy the restriction property. Instead, it satisfies h-longevity
whenever $a_{u}>0$ is increasing in $u$.

b) Consider now the following driver: $g_{u}(t,z)=bz+a_{u}$ for any
$t\in[0,u],z\in\mathbb{R}^{d}$, with $a_{u},b\in\mathbb{R}\setminus\\{0\\}$
and $a_{u}$ depending on the maturity $u$. Similarly to Example 10, it follows
that

$\rho_{tu}(X)=E_{Q}\left[\left.-X\right|\mathcal{F}_{t}\right]+(u-t)a_{u},$

where
$E\left[\left.\frac{dQ}{dP}\right|\mathcal{F}_{t}\right]=\exp\left\\{-\frac{1}{2}b^{2}t+b\cdot
B_{t}\right\\}$. As before, therefore, for $a_{u}\neq 0$, $(\rho_{tu})_{t,u}$
is not normalized and does not satisfy the restriction property. Instead, it
satisfies h-longevity whenever $a_{u}>0$ is increasing in $u$.

####  3.2.1 Time-consistency

Thanks to the Comparison Theorem (see El Karoui et al. [13]) and to the
Converse Comparison Theorem (see Briand et al. [8]), the following result
establishes that monotonicity of the family $\mathcal{G}=(g_{t})_{t\in[0,T]}$
is equivalent to sub time-consistency.

###### Theorem 15

Let $(\rho_{tu})_{t,u}$ be induced by a BSDE with a family of drivers
$\mathcal{G}=(g_{t})_{t\in[0,T]}$ as in (33).

a) The family $\mathcal{G}$ is increasing if and only if $(\rho_{tu})_{t,u}$
satisfies sub time-consistency.

b) $\mathcal{G}=\\{g\\}$ if and only if $(\rho_{tu})_{t,u}$ satisfies strong
time-consistency.

Proof. a) By definition (33), sub time-consistency of $(\rho_{tu})_{t,u}$ can
be written as

$\mathcal{E}^{g_{t}}(\mathcal{E}^{g_{u}}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})\leq\mathcal{E}^{g_{u}}(-X|\mathcal{F}_{s})$
(34)

for any $s\leq t\leq u$ and $X\in L^{2}(\mathcal{F}_{u})$.

By definition of $\mathcal{E}^{g_{t}}$, the left-hand and right-hand sides of
the previous equation can be rewritten as follows:

$\displaystyle\mathcal{E}^{g_{t}}(\mathcal{E}^{g_{u}}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})$
$\displaystyle=$
$\displaystyle\mathcal{E}^{g_{u}}(-X|\mathcal{F}_{t})+\int_{s}^{t}g_{t}(v,Z_{v})\,dv-\int_{s}^{t}Z_{v}\,dB_{v}$
$\displaystyle=$
$\displaystyle-X+\int_{t}^{u}g_{u}(v,\tilde{Z}_{v})\,dv-\int_{t}^{u}\tilde{Z}_{v}\,dB_{v}+\int_{s}^{t}g_{t}(v,Z_{v})\,dv-\int_{s}^{t}Z_{v}\,dB_{v}$

and

$\mathcal{E}^{g_{u}}(-X|\mathcal{F}_{s})=-X+\int_{s}^{u}g_{u}(v,\hat{Z}_{v})\,dv-\int_{s}^{u}\hat{Z}_{v}\,dB_{v}.$
(35)

Set now

$\bar{g}(v,z)=\left\\{\begin{array}[]{rl}g_{t}(v,z);&\quad v\in[0,t]\\\
g_{u}(v,z);&\quad
v\in(t,u]\end{array}\right.\quad\bar{Z}_{v}=\left\\{\begin{array}[]{rl}Z_{v};&\quad
v\in[0,t]\\\ \tilde{Z}_{v};&\quad v\in(t,u]\end{array}\right.$

By the arguments above, the left-hand side of (34) becomes

$\mathcal{E}^{g_{t}}(\mathcal{E}^{g_{u}}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})=-X+\int_{s}^{u}\bar{g}(v,\bar{Z}_{v})\,dv-\int_{s}^{u}\bar{Z}_{v}\,dB_{v}.$
(36)

On the one hand, if $\bar{g}\leq g_{u}$ on $[0,u]$ (or, equivalently,
$g_{t}\leq g_{u}$ on $[0,t]$), then by the Comparison theorem of BSDE (see El
Karoui et al. [13]) sub time-consistency (34) is satisfied.

On the other hand, by (35) and (36) and Converse Comparison Theorem of Briand
et al. [8], sub time-consistency (34) implies that $\bar{g}\leq g_{u}$ on
$[0,u]$, hence $g_{t}\leq g_{u}$ on $[0,t]$.

b) The case of time-consistency can be obtained by replacing inequalities with
equalities in the proof above.

In other words, the result above guarantees that strong time-consistency is
fulfilled if and only if the family of drivers reduces to a singleton, that
is, the drivers $g_{t}$ do not depend on the maturity $t$.

###### Remark 16

In the previous result, the proof of item a) can be done also by means of the
penalty term characterization. In this case, one could use

$\alpha_{su}^{\mathcal{G}}(Q)=E_{Q}\left[\left.\int_{s}^{u}g^{*}_{u}(v,q_{v})dv\right|\mathcal{F}_{s}\right]$

and the fact that, by the Fenchel-Moreau biconjugate Theorem, the condition
$g^{*}_{t}\geq g^{*}_{u}$ is equivalent to $g_{t}\leq g_{u}$. Indeed, if
$g^{*}_{t}\geq g^{*}_{u}$ then

$g_{t}(s,z)=\sup_{q\in\mathbb{R}^{d}}\\{q\cdot
z-g_{t}^{*}(s,q)\\}\leq\sup_{q\in\mathbb{R}^{d}}\\{q\cdot
z-g_{u}^{*}(s,q)\\}=g_{u}(s,z).$

The converse implication can be checked similarly.

####  3.2.2 H-longevity and families of BSDEs

In order to investigate h-longevity, we need to compare BSDEs with different
horizons. Hereafter, we suggest a comparison theorem for this. Note that in
this result we consider more general BSDEs (28) with driver depending on $y$,
as the result is of interest well beyond the use of BSDEs in risk measures
modelling.

Let $T_{1},T_{2}>0$ be two different time horizon with $T_{2}>T_{1}$. Let
$(Y_{t}^{T_{i},\xi_{i}},Z_{t}^{T_{i},\xi_{i}})_{t\in[0,T_{1}]}$ (for $i=1,2$)
be the solution of a BSDE with final condition $\xi_{i}\in
L^{2}(\mathcal{F}_{T_{i}})$ and driver $g^{T_{i}}$ satisfying the usual
assumptions. More precisely,

$Y_{t}^{T_{i}}=\xi_{i}+\int_{t}^{T_{i}}g^{T_{i}}(s,Y_{s}^{T_{i}},Z_{s}^{T_{i}})ds-\int_{t}^{T_{i}}Z_{s}^{T_{i}}dB_{s}.$
(37)

In the terminology of Peng [21],
$Y_{t}^{T_{i}}=\mathcal{E}^{g^{T_{i}}}(\xi_{i}\big{|}\mathcal{F}_{t})$.

###### Theorem 17

(Horizon Comparison Theorem) If $g^{T_{2}}(s,y,z)\geq g^{T_{1}}(s,y,z)$ for
any $s\in[0,T_{1}],y\in\mathbb{R},z\in\mathbb{R}^{d}$ and
$g^{T_{2}}(s,y,z)\geq 0$ for any
$s\in[T_{1},T_{2}],y\in\mathbb{R},z\in\mathbb{R}^{d}$ and
$\xi_{2}\geq\xi_{1}$, then $Y_{t}^{T_{2}}\geq Y_{t}^{T_{1}}$ for any
$t\in[0,T_{1}]$ and $Y_{t}^{T_{2}}\geq\xi_{1}$ for any $t\in[T_{1},T_{2}]$.

Proof. By (37), it follows that

$\displaystyle Y_{t}^{T_{2}}-Y_{t}^{T_{1}}=$
$\displaystyle\xi_{2}-\xi_{1}+\int_{t}^{T_{2}}g^{T_{2}}(s,Y_{s}^{T_{2}},Z_{s}^{T_{2}})ds-\int_{t}^{T_{1}}g^{T_{1}}(s,Y_{s}^{T_{1}},Z_{s}^{T_{1}})ds$
(38)
$\displaystyle-\int_{t}^{T_{2}}Z_{s}^{T_{2}}dB_{s}+\int_{t}^{T_{1}}Z_{s}^{T_{1}}dB_{s}.$

Let us now extend $g^{T_{1}}(s,y,z)$, $Y_{s}^{T_{1}}$ and $Z_{s}^{T_{1}}$ also
to $[T_{1},T_{2}]$ as follows:

$\displaystyle\overline{g}^{(T_{1})}(s,y,z)=\left\\{\begin{array}[]{rl}g^{T_{1}}(s,y,z);&s\in[0,T_{1}]\\\
0;&s\in(T_{1};T_{2}]\end{array}\right.$
$\displaystyle\overline{Y}_{s}^{T_{1}}=\left\\{\begin{array}[]{rl}Y_{s}^{T_{1}};&s\in[0,T_{1}]\\\
\xi_{1};&s\in(T_{1};T_{2}]\end{array}\right.;\qquad\overline{Z}_{s}^{T_{1}}=\left\\{\begin{array}[]{rl}Z_{s}^{T_{1}};&s\in[0,T_{1}]\\\
0;&s\in(T_{1};T_{2}]\end{array}\right.$

For $t\in[0,T_{2}]$, equation (38) becomes then

$Y_{t}^{T_{2}}-\overline{Y}_{t}^{T_{1}}=\xi_{2}-\xi_{1}+\int_{t}^{T_{2}}\big{[}g^{T_{2}}(s,Y_{s}^{T_{2}},Z_{s}^{T_{2}})-\overline{g}^{T_{1}}(s,\overline{Y}_{s}^{T_{1}},\overline{Z}_{s}^{T_{1}})\big{]}ds\\\
-\int_{t}^{T_{2}}(Z_{s}^{T_{2}}-\overline{Z}_{s}^{T_{1}})dB_{s}$ (41)

Set now $\widetilde{Y}_{s}\triangleq Y_{s}^{T_{2}}-\overline{Y}_{s}^{T_{1}}$,
$\widetilde{\xi}\triangleq\xi_{2}-\xi_{1}$, $\widetilde{Z}^{i}_{s}\triangleq
Z_{s}^{T_{2},i}-\overline{Z}_{s}^{T_{1},i}$ for $i=1,...,d$, and

$\displaystyle\Delta_{s}g$ $\displaystyle=$ $\displaystyle
g^{T_{2}}(s,\overline{Y}_{s}^{T_{1}},\overline{Z}_{s}^{T_{1}})-\overline{g}^{T_{1}}(s,\overline{Y}_{s}^{T_{1}},\overline{Z}_{s}^{T_{1}})$
$\displaystyle\Delta_{s}y$ $\displaystyle=$
$\displaystyle\frac{g^{T_{2}}(s,Y_{s}^{T_{2}},Z_{s}^{T_{2}})-g^{T_{2}}(s,\overline{Y}_{s}^{T_{1}},Z_{s}^{T_{2}})}{Y_{s}^{T_{2}}-\overline{Y}_{s}^{T_{1}}}1_{\\{Y_{s}^{T_{2}}\neq\overline{Y}_{s}^{T_{1}}\\}}$
$\displaystyle(\Delta_{s}z)^{i}$ $\displaystyle=$
$\displaystyle\frac{g^{T_{2}}(s,\overline{Y}_{s}^{T_{1}},Z_{s}^{T_{2}})-g^{T_{2}}(s,\overline{Y}_{s}^{T_{1}},\overline{Z}_{s}^{T_{1}})}{d(Z_{s}^{T_{2},i}-\overline{Z}_{s}^{T_{1},i})}1_{\\{Z_{s}^{T_{2},i}\neq\overline{Z}_{s}^{T_{1},i}\\}}$

for $i=1,...,d$. Hence (41) can be rewritten as

$\left\\{\begin{array}[]{rl}-d\widetilde{Y}_{t}&=\left[\Delta_{t}y\,\widetilde{Y}_{t}+\Delta_{t}z\cdot\widetilde{Z}_{t}+\Delta_{t}\,g\right]dt-\widetilde{Z}_{t}dB_{t}\\\
\widetilde{Y}_{T_{2}}&=\widetilde{\xi}\end{array}\right.$ (42)

Since
$\Delta_{t}y\in\mathbb{H}^{2}_{[0,T]}(\mathbb{R}),\Delta_{t}z\in\mathbb{H}^{2}_{[0,T]}(\mathbb{R}^{d})$
and $\Delta_{t}\,g$ is in $\mathbb{H}^{2}_{[0,T]}(\mathbb{R})$ by the
assumption of $g^{T_{i}}$ Lipschitz, the solution of (42) exists uniquely by
Prop. 2.2 of El Karoui, Peng and Quenez [13]. Moreover, since
$g^{T_{2}}(s,y,z)\geq\overline{g}^{T_{1}}(s,y,z)$ for any $s\in[0,T_{2}],y,z$
and $\widetilde{\xi}\geq 0$, then $\widetilde{Y}_{t}\geq 0$ for any
$t\in[0,T_{2}]$ which concludes the proof.

The previous result guarantees that for fully-dynamic risk measures induced by
$g$-expectations as in (33), h-longevity holds when $g_{u}$ is increasing in
$u$ and $g_{u}\geq 0$.

###### Proposition 18

a) If $\mathcal{G}$ is an increasing family of drivers and $\rho_{tu}(0)\geq
0$ for any $t\leq u$, then $(\rho_{tu})_{t,u}$ satisfies h-longevity.

b) If $\mathcal{G}$ is an increasing family of drivers and $g_{t}\geq 0$ for
any $t\in[0,T]$, then $(\rho_{tu})_{t,u}$ satisfies h-longevity.

Proof. a) follows by Remark 6 and Proposition 15.

b) By Theorem 17, increasing monotonicity of the family $\mathcal{G}$ and
$g_{t}\geq 0$ for any $t\in[0,T]$ imply h-longevity.

Notice that requiring $g_{u}\geq 0$ implies that $\rho_{tu}(X)\geq
E_{P}[-X|\mathcal{F}_{t}]$ for any $X\in L^{2}(\mathcal{F}_{u})$. Indeed, by
definition of $\rho_{tu}(X)$ as in (33) and from $g_{u}\geq 0$, it follows
that

$\displaystyle\rho_{tu}(X)$ $\displaystyle=$ $\displaystyle
E_{P}\left[\left.-X+\int_{t}^{u}g_{u}(v,Z_{v})\,dv-\int_{t}^{u}Z_{v}\,dB_{v}\right|\mathcal{F}_{t}\right]$
$\displaystyle=$ $\displaystyle
E_{P}\left[-X|\mathcal{F}_{t}\right]+E_{P}\left[\left.\int_{t}^{u}g_{u}(v,Z_{v})\,dv\right|\mathcal{F}_{t}\right]$
$\displaystyle\geq$ $\displaystyle E_{P}\left[-X|\mathcal{F}_{t}\right].$

####  3.2.3 Beyond the Lipschitz driver: the entropic case

Next, we consider the entropic risk measure, which is generated by a quadratic
BSDE. Even though the Lipschitz condition is clearly not satisfied in this
case, the solution of the BSDE is still unique. This discussion on the
uniqueness of solutions is developed in [3]. The entropic risk measure is
particularly interesting because of its explicit connection with utility
functions as investment preference criteria. In the following example, we
propose a variation to the classical entropic risk measure able to capture
h-longevity.

###### Example 19

In the one-dimensional case, for every $u\in[0,T]$, consider the driver
$g_{u}(t,z)=\frac{b_{u}}{2}z^{2}+a_{u}(t)$ for any $t\in[0,T],z\in\mathbb{R}$,
with $b_{u}>0$ and $a_{u}$ positive function. We recall that $b_{u}$ is the
reciprocal of the risk aversion parameter in the corresponding utility
function. By using the explicit solution of the BSDE with driver
$\bar{g}_{u}(t,z)=\frac{b_{u}}{2}z^{2}$, it can be checked easily that, for
any $t\in[0,T]$ and $X\in L^{2}(\mathcal{F}_{u})$,

$\rho_{su}(X)=\frac{1}{b_{u}}\ln\left(E_{P}\left[\left.\exp(-b_{u}X)\right|\mathcal{F}_{s}\right]\right)+\int_{s}^{u}a_{u}(t)dt.$

Hence, $(\rho_{su})_{s,u}$ is not normalized and does not satisfy the
restriction property. Instead,  (strict) h-longevity holds whenever
$(b_{u})_{u}$ and $(a_{u})_{u}$ are increasing in $u$. See Proposition 18.

##  4 Relation with BSVIEs

We have come now to consider a fully-dynamic risk measure induced by a single
BSVIE with a driver $g$ and also measures generated by a family of BSVIEs
associated to the drivers $\mathcal{G}=(g_{t})_{t\in[0,T]}$ where the index
$t$ refers to the time horizon.

###  4.1 Risk measures generated by a single BSVIE

In the same spirit of risk measures generated by BSDEs, we can consider risk
measures induced by BSVIEs as Agram [2] and Yong [29] suggested. Indeed, this
can also be seen in connection with families of BSDEs as Yong [30] has
pinpointed.

Let us consider a BSVIE of type

$Y_{t}=\xi_{t}+\int_{t}^{T}g(t,s,Z(t,s))\,ds-\int_{t}^{T}Z(t,s)\,dB_{s},$ (43)

where $\xi_{t}\in L^{2}(\mathcal{F}_{T})$, for all $t\in[0,T]$. The driver
$g:\Omega\times\Delta\times\mathbb{R}^{d}\to\mathbb{R}$, with
$\Delta_{[0,T]}\triangleq\\{(t,s)\in[0,T]\times[0,T]:s\geq t\\}$, is
independent of $y$ (so to guarantee translation invariance in the application
to monetary risk measures) and it is such that

  * •

for any $t\in[0,T],z\in\mathbb{R}^{d}$, $g(t,\cdot,z)$ is adapted;

  * •

is uniformly Lipschitz, i.e. there exists a constant $C>0$ such that

$|g(t,s,z_{1})-g(t,s,z_{2})|\leq C|z_{1}-z_{2}|,\mbox{ for any
}(t,s)\in\Delta,z_{1},z_{2}\in\mathbb{R}^{d},P-a.s.;$

  * •

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

Under there assumptions, we have a unique solution
$(Y_{\cdot},Z(\cdot,\cdot))\in\mathbb{H}^{2}_{[0,T]}(\mathbb{R})\times\mathcal{Z}^{2}_{[0,T]}(\mathbb{R}^{d})$,
where we have set

$\mathcal{Z}^{2}_{[a,b]}(\mathbb{R}^{d})\triangleq\left\\{Z:\Omega\times\Delta_{[a,b]}\to\mathbb{R}^{d}:\begin{array}[]{rl}&Z(t,\cdot)\mbox{
is adapted for all }t\mbox{ and }\\\
&E\left[\int_{a}^{b}\int_{a}^{b}|Z(t,s)|^{2}\,ds\,dt\right]<+\infty\end{array}\right\\}.$

See Yong [30], Thm. 2.2, and [29]. Also Yong [30] introduced the following
family of BSDEs parameterized by $t$ and related to the BSVIE (43):

$\eta(r;t,\xi_{t})=\xi_{t}+\int_{r}^{T}\bar{g}(v,\zeta(v;t);t)dv-\int_{r}^{T}\zeta(v;t)dB_{v},\quad
r\in[t,T]$

where

$\zeta(v;t)=Z(t,v);\quad\bar{g}(v,\zeta(v;t);t)=g(t,v,Z(t,v))\quad\mbox{and}\quad
Y_{t}=\eta(t;t,\xi_{t}).$

Although $\eta(\cdot;t,X)$ and $Y_{t}$ are closely related, $Y_{t}$, solution
of the BSVIE (43) may satisfy some specific properties, since it corresponds
not to the whole family $\eta(\cdot;t,X)$, but only to $\eta(t;t,X)$.

As in Section 3, we assume convex drivers that guarantee convex risk measures
satisfying monotonicity and translation invariance. See Yong [29] and Agram
[2], who also makes an extension to jump dynamics. Then we focus on monetary
risk measures of the type

$\rho_{tu}(X)=\mathcal{E}^{g,V}\left(-X|\mathcal{F}_{t}\right),\quad X\in
L^{2}(\mathcal{F}_{u}),$ (44)

where $\mathcal{E}^{g,V}\left(-X|\mathcal{F}_{t}\right)$ denotes the
$Y$-component of the solution of the BSVIE of type (43), with terminal
condition $\xi_{t}=-X$, for all $t$.

As shown in the following result, for a driver independent on $y$ the
condition $g(t,u,0)=0$, for any $t\leq u$, guarantees the restriction property
as well as the normalization. Hence, in our case we cannot assume
$g(t,u,0)=0$.

###### Proposition 20

The following statements are equivalent:

(a) $g(t,u,0)=0$ for any $0\leq t\leq u\leq T$;

(b) $\rho$ is normalized, see (3):

(c) $\rho$ satisfies the restriction property (2).

Proof. (a) $\Rightarrow$ (b) The implication is true because
$(Y_{\cdot},,Z(\cdot,\cdot))$ with $Y_{t}=Z(t,s)=0$, $s\geq t$ is the unique
solution corresponding to $X=0$.

(b) $\Rightarrow$ (a). Assume that $\rho_{tu}(0)=0$ for any $0\leq t\leq u\leq
T$. we have

$0=Y_{t}=\int_{t}^{u}g(t,s,Z(t,s))ds-\int_{t}^{u}Z(t,s)dB_{s}$

holds for any $0\leq t\leq u\leq T$. Hence

$\int_{t}^{u}g(t,s,Z(t,s))ds=\int_{t}^{u}Z(t,s)dB_{s},\quad\mbox{ for any
}0\leq t\leq u\leq T.$

As in the proof of Proposition 9, it follows that for any fixed $t$ the
martingale $M_{t}(u)=\int_{t}^{u}Z(t,s)dB_{s}$, with $u\geq t$, starting from
$0$ should be identically to $0$. Hence $Z(t,s)\equiv 0$ for any $s\geq t$ and
$\int_{t}^{u}g(t,s,0)ds\equiv 0$ for any $u\geq t$. By deriving this last
equation with respect to $u$, it follows that $g(t,u,0)=0$ for any $u\geq t$.

(a) $\Rightarrow$ (c). Assume that $X\in L^{2}(\mathcal{F}_{u})$ and consider
$\rho_{tu}(X)$ and $\rho_{tv}(X)$ for any $v\geq u$. Let us denote by
$(Y^{X,u}_{r},Z^{X,u}(r,s))$ (resp. $(Y^{X,v}_{r},Z^{X,v}(r,s))$) the solution
corresponding to $\rho_{tu}(X)$ (resp. $\rho_{tv}(X)$) at time $r\leq u$.
Since $(Y^{X,v}_{r},Z^{X,v}(r,s))$ with

$Y^{X,v}_{r}=\left\\{\begin{array}[]{rl}Y^{X,u}_{r};&r\leq u\\\ -X;&u<r\leq
v\end{array}\right.\quad
Z^{X,v}(r,s)=\left\\{\begin{array}[]{rl}Z^{X,u}(r,s);&s\leq u\\\ 0;&u<s\leq
v\end{array}\right.$

is a solution of $\rho_{tv}(X)$ whether $g(t,u,0)=0$ for any $0\leq t\leq
u\leq T$, the restriction property follows.

(c) $\Rightarrow$ (a) The argument is similar to the proof of the
corresponding implication in Proposition 9.

Note that, for all $t$, the set of probability measures $Q_{t}$ on
$\mathcal{F}_{u}$ such that

$\frac{dQ_{t}}{dP}\triangleq\exp\left\\{\frac{1}{2}\int_{t}^{u}|q(t,s)|^{2}ds-\int_{t}^{u}q(t,s)dB_{s}\right\\},$
(45)

for some stochastic process $(q(t,s))_{t\leq s\leq
u}\in\mathbb{H}^{2}_{[t,u]}(\mathbb{R}^{d})$ coincides with $\mathcal{Q}_{tu}$
in a Brownian setting (see Revuz and Yor [24], Chapter VIII, Prop. 1.6).

###### Theorem 21

(Dual representation) For any $t\leq u$ fixed, the risk measure $\rho_{tu}$
induced by a BSVIE as in (44) has the following dual representation

$\rho_{tu}(X)=\underset{Q_{t}\in\mathcal{Q}_{tu}}{\operatorname{ess.\\!sup}}\left\\{E_{Q_{t}}\left[-X|\mathcal{F}_{t}\right]-\alpha_{tu}(Q_{t})\right\\},$
(46)

where $Q_{t}$ corresponds to $(q(t,s))_{t\leq s\leq u}$ via (45) and the
penalty functional is given by

$\alpha_{tu}(Q_{t})=E_{Q_{t}}\left[\left.\int_{t}^{u}g^{*}(t,s,q(t,s))ds\right|\mathcal{F}_{t}\right],$

with $g^{*}(t,s,\cdot)$ denoting the convex conjugate of $g(t,s,\cdot)$.

Proof. The present proof extends a similar one of Barrieu and El Karoui [3] to
the case of BSVIEs. Let $t,u\in[0,T]$ with $t\leq u$ be fixed arbitrarily. The
evaluation of risk (44) can be written as

$\displaystyle Y_{t}$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{u}[g(t,s,Z(t,s))-q(t,s)\cdot
Z(t,s)]ds-\int_{t}^{u}Z(t,s)(dB_{s}-q(t,s)ds)$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{u}[g(t,s,Z(t,s))-q(t,s)\cdot
Z(t,s)]ds-\int_{t}^{u}Z(t,s)dB^{Q_{t}}_{s}$

and, by Girsanov Theorem, $B^{Q_{t}}_{u}\triangleq
B_{u}-B_{t}-\int_{t}^{u}q(t,s)ds$, $u\geq t$, is a Brownian motion with
respect to $Q_{t}$ (see (45)) with initial value $B^{Q_{t}}_{t}=0$. Set now

$g^{*}(t,s,q)\triangleq\underset{z\in\mathbb{R}^{d}}{\operatorname{ess.\\!sup}}\\{q\cdot
z-g(t,s,z)\\},\,q\in\mathbb{R}^{d},$

for $s,t$ such that $s\geq t$. We obtain that, for any fixed $t$,
$g^{*}(t,\cdot,q(t,\cdot))$ is adapted. Furthermore,

$\displaystyle Y_{t}$ $\displaystyle=$
$\displaystyle-X-\int_{t}^{u}[q(t,s)\cdot
Z(t,s)-g(t,s,Z(t,s))]ds-\int_{t}^{u}Z(t,s)dB^{Q_{t}}_{s}$ $\displaystyle\geq$
$\displaystyle-X-\int_{t}^{u}g^{*}(t,s,q(t,s))ds-\int_{t}^{u}Z(t,s)dB^{Q_{t}}_{s},$

where the last inequality follows by the definition of $g^{*}$. By taking the
conditional expectation with respect to $Q_{t}$, it follows

$Y_{t}\geq
E_{Q_{t}}\left[-X|\mathcal{F}_{t}\right]-E_{Q_{t}}\left[\left.\int_{t}^{u}g^{*}(t,s,q(t,s))ds\right|\mathcal{F}_{t}\right],$

since $E_{Q_{t}}[\int_{t}^{u}Z(t,s)dB^{Q_{t}}_{s}|\mathcal{F}_{t}]=0$ because
$(B^{Q_{t}}_{s})_{s\geq t}$ is a Brownian motion with respect to $Q_{t}$.
Consequently,

$Y_{t}\geq\underset{Q_{t}\in\mathcal{Q}_{tu}}{\operatorname{ess.\\!sup}}\Big{\\{}E_{Q_{t}}\big{[}-X|\mathcal{F}_{t}\big{]}-E_{Q_{t}}\Big{[}\int_{t}^{u}g^{*}(t,s,q(t,s))ds\big{|}\mathcal{F}_{t}\Big{]}\Big{\\}}.$
(47)

It remains to prove the converse inequality in (47). In a similar way as in
Theorem 7.4 (i) and Lemma 7.5 of Barrieu and El Karoui [3], for $t$ fixed and
any $Z(t,s)$, $s\geq t$, there exist some progressively measurable
$q_{Z}(t,s)$ (associated to a $Q_{t,Z}$ via (45)) such that
$g(t,s,Z(t,s))=q_{Z}(t,s)\cdot Z(t,s)-g^{*}(t,s,q_{Z}(t,s))$. Hence,

$\displaystyle Y_{t}$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{u}g(t,s,Z(t,s))ds-\int_{t}^{u}Z(t,s)dB_{s}$
$\displaystyle=$ $\displaystyle-X+\int_{t}^{u}[q_{Z}(t,s)\cdot
Z(t,s)-g^{*}(t,s,q_{Z}(t,s))]ds-\int_{t}^{u}Z(t,s)dB_{s}$ $\displaystyle=$
$\displaystyle-X-\int_{t}^{u}g^{*}(t,s,q_{Z}(t,s))ds-\int_{t}^{u}Z(t,s)dB^{Q_{t,Z}}_{s},$

where $B^{Q_{t,Z}}_{s}=B_{s}-B_{t}-\int_{t}^{s}q_{Z}(t,r)dr$, $t\leq s\leq u$.
By taking the conditional expectation with respect to $Q_{t,Z}$, it holds that

$\displaystyle Y_{t}$ $\displaystyle=$ $\displaystyle
E_{Q_{t,Z}}\left[\left.-X-\int_{t}^{u}g^{*}(t,s,q_{Z}(t,s))ds\right|\mathcal{F}_{t}\right]$
(48) $\displaystyle\leq$
$\displaystyle\underset{Q_{t}\in\mathcal{Q}_{tu}}{\operatorname{ess.\\!sup}}\left\\{E_{Q_{t}}\left[-X|\mathcal{F}_{t}\right]-E_{Q_{t}}\left[\left.\int_{t}^{u}g^{*}(t,s,q(t,s))ds\right|\mathcal{F}_{t}\right]\right\\}.$

The thesis follows then by (47) and (48).

####  4.1.1 Time-consistency

The result below provides a necessary and sufficient condition on $g$ for sub
time-consistency of $(\rho_{tu})_{t,u}$ induced by a BSVIE $(g)$.

###### Theorem 22

Let $(\rho_{tu})_{t,u}$ be induced by a BSVIE with driver $g$ as in (44).

a) The driver $g(t,v,z)$ is decreasing in $t$ for any
$v\in[t,u],z\in\mathbb{R}^{d}$ (meaning that, for any $s\leq t$, $g(t,v,z)\leq
g(s,v,z)$) if and only if $(\rho_{tu})_{t,u}$ satisfies sub time-consistency.

b) If the driver $g(t,\cdot,\cdot)$ is constant in $t$, then
$(\rho_{tu})_{t,u}$ satisfies time-consistency.

Proof. a) Assume that sub time-consistency holds. By the dual representation
of $(\rho_{tu})_{t,u}$ in Proposition 21, the penalty term of $\rho_{tu}$ is
given by

$\alpha_{tu}(Q_{t})=E_{Q_{t}}\left[\left.\int_{t}^{u}g^{*}(t,v,q(t,v))dv\right|\mathcal{F}_{t}\right]$

for any $Q_{t}\in\mathcal{Q}_{tu}$. Let $s,t,u\in[0,T]$ with $s\leq t\leq u$
and let $Q_{s}\in\mathcal{Q}_{st},Q_{t}\in\mathcal{Q}_{tu}$ be fixed
arbitrarily. Set now $\bar{Q}=Q_{s}\cdot Q_{t}$ the pasting of $Q_{s}$ on
$[s,t]$ and of $Q_{t}$ on $[t,u]$, hence $\bar{Q}\in\mathcal{Q}_{su}$. Denote
by $q(s,v)$, $q(t,v)$ and $\bar{q}(s,v)$ the corresponding processes as in
(45). By applying the characterization of penalty term of sub time-consistent
risk measures (see Proposition 7(iii)), we obtain

$\displaystyle{\displaystyle
E_{\bar{Q}}\Big{[}\int_{s}^{u}g^{*}(s,v,\bar{q}(s,v))dv\Big{|}\mathcal{F}_{s}\Big{]}}$
$\displaystyle{\displaystyle\leq
E_{\bar{Q}}\Big{[}\int_{s}^{t}g^{*}(s,v,\bar{q}(s,v))dv\Big{|}\mathcal{F}_{s}\Big{]}+E_{\bar{Q}}\Big{[}E_{\bar{Q}}\Big{[}\int_{t}^{u}g^{*}(t,v,\bar{q}(t,v))dv\Big{|}\mathcal{F}_{t}\Big{]}\Big{|}\mathcal{F}_{s}\Big{]}},$

hence

$\displaystyle{\displaystyle
E_{\bar{Q}}\Big{[}\int_{t}^{u}g^{*}(s,v,\bar{q}(s,v))dv\Big{|}\mathcal{F}_{s}\Big{]}\leq
E_{\bar{Q}}\Big{[}E_{\bar{Q}}\Big{[}\int_{t}^{u}g^{*}(t,v,\bar{q}(t,v))dv\Big{|}\mathcal{F}_{t}\Big{]}\Big{|}\mathcal{F}_{s}\Big{]}}$
$\displaystyle{\displaystyle
E_{Q_{s}}\Big{[}E_{Q_{t}}\Big{[}\int_{t}^{u}g^{*}(s,v,\bar{q}(s,v))dv\Big{|}\mathcal{F}_{t}\Big{]}\Big{|}\mathcal{F}_{s}\Big{]}\leq
E_{Q_{s}}\Big{[}E_{Q_{t}}\Big{[}\int_{t}^{u}g^{*}(t,v,\bar{q}(t,v))dv\Big{|}\mathcal{F}_{t}\Big{]}\Big{|}\mathcal{F}_{s}\Big{]}}$
$\displaystyle{\displaystyle
E_{Q_{s}}\Big{[}E_{Q_{t}}\Big{[}\int_{t}^{u}\Big{[}g^{*}(t,v,\bar{q}(t,v))-g^{*}(s,v,\bar{q}(s,v))\Big{]}dv\Big{|}\mathcal{F}_{t}\Big{]}\Big{|}\mathcal{F}_{s}\Big{]}\geq
0.}$ (49)

Since (49) should hold for any $s\leq t\leq u$ and any
$Q_{s}\in\mathcal{Q}_{st},Q_{t}\in\mathcal{Q}_{tu}$, it follows that

$g^{*}(t,v,\bar{q}(t,v))\geq g^{*}(s,v,\bar{q}(s,v))\quad\mbox{ for any }s\leq
t,v\in[t,u],\,\bar{q}\in\mathbb{R}^{d}.$

Hence, $g(t,\cdot,\cdot)\leq g(s,\cdot,\cdot)$ for any $s\leq t$.

Conversely, assume that $g(t,\cdot,\cdot)$ is decreasing in $t$, in the
meaning specified above. We are going to prove that

$\rho_{st}(-\rho_{tu}(X))=\mathcal{E}^{g,V}(\mathcal{E}^{g,V}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})\leq\mathcal{E}^{g,V}(-X|\mathcal{F}_{s})=\rho_{su}(X)$
(50)

for any $s\leq t\leq u$ and $X\in L^{2}(\mathcal{F}_{u})$. By definition of
$\mathcal{E}^{g,V}$, the left-hand and right-hand sides of the previous
equation can be rewritten as follows:

$\displaystyle\mathcal{E}^{g,V}(\mathcal{E}^{g,V}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})$
$\displaystyle=$
$\displaystyle\mathcal{E}^{g,V}(-X|\mathcal{F}_{t})+\int_{s}^{t}g(s,v,Z(s,v))\,dv-\int_{s}^{t}Z(s,v)\,dB_{v}$
$\displaystyle=$
$\displaystyle-X+\int_{t}^{u}g(t,v,\tilde{Z}(t,v)\,dv-\int_{t}^{u}\tilde{Z}(t,v)\,dB_{v}+\int_{s}^{t}g(s,v,Z(s,v))\,dv-\int_{s}^{t}Z(s,v)\,dB_{v}$
$\displaystyle=$
$\displaystyle-X+\int_{s}^{u}\Big{[}g(s,v,Z(s,v))1_{[s,t]}(v)+g(t,v,\tilde{Z}(t,v))1_{(t,u]}(v)\Big{]}\,dv$
(51)
$\displaystyle-\int_{s}^{u}\big{[}Z(s,v)1_{[s,t]}(v)+\tilde{Z}(t,v)1_{(t,u]}(v)\big{]}\,dB_{v}$

and

$\mathcal{E}^{g,V}(-X|\mathcal{F}_{s})=-X+\int_{s}^{u}g(s,v,\hat{Z}(s,v)\,dv-\int_{s}^{u}\hat{Z}(s,v)\,dB_{v}.$
(52)

Furthermore, by decreasing monotonicity of $g(t,\cdot,\cdot)$ in $t$ and by
(51) it follows that

$\displaystyle\mathcal{E}^{g,V}(\mathcal{E}^{g,V}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})$
$\displaystyle\leq$
$\displaystyle-X+\int_{s}^{u}\left[g(s,v,Z(s,v))1_{[s,t]}(v)+g(s,v,\tilde{Z}(t,v))1_{(t,u]}(v)\right]\,dv$
$\displaystyle-\int_{s}^{u}\left[Z(s,v)1_{[s,t]}(v)+\tilde{Z}(t,v)1_{(t,u]}(v)\right]\,dB_{v}.$

Set now

$\bar{Z}_{t}(s,v)=Z(s,v)1_{[s,t]}(v)+\tilde{Z}(t,v)1_{(t,u]}(v),\quad v\geq
s.$

By the arguments above,

$\mathcal{E}^{g,V}(\mathcal{E}^{g,V}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})\leq-X+\int_{s}^{u}g(s,v,\bar{Z}_{t}(s,v)\,dv-\int_{s}^{u}\bar{Z}_{t}(s,v)\,dB_{v}.$
(53)

Sub time-consistency (50) then follows by comparing (52) and (53) and by the
uniqueness of the solution of a BSVIE.

b) The case of time-consistency can be obtained by replacing inequalities with
equalities in the proof above.

Notice that a BSVIE of the form (52), that is with a final condition $X$ that
does not depend on $t$ and with a driver $g$ independent of $y$, reduces to a
BSDE when the driver $g(t,v,z)$ is constant in $t$.

An alternative proof of item a) could be given in terms of a Converse
Comparison Theorem for BSVIEs (similarly to Theorem 4.1 of Briand et al.[8]
for BSDEs) if that result was available. However, while for BSDEs the Converse
Comparison Theorem has been proved by Briand et al. [8], for BSVIEs we are not
aware of any similar result. We prove below that a Converse Comparison Theorem
holds also for BSVIEs.

We recall that a BSVIE (43) is related to the following family of BSDEs
parameterized by $t$:

$\eta(r;t,\xi_{t})=\xi_{t}+\int_{r}^{T}\bar{g}(v,\zeta(v;t);t)dv-\int_{r}^{T}\zeta(v;t)dB_{v},\quad
r\in[t,T]$ (54)

where

$\zeta(v;t)=Z(t,v);\quad\bar{g}(v,\zeta(v;t);t)=g(t,v,Z(t,v))\quad\mbox{and}\quad
Y_{t}=\eta(t;t,\xi_{t}).$

We remark again that the drivers above do not depend on $y$. In this way,
translation invariance of the associated risk measure is not guaranteed. We
are then able to prove a Converse Comparison Theorem for BSVIEs.

###### Theorem 23

(Converse Comparison Theorem for BSVIEs) Let $Y_{t}^{1}$ and $Y_{t}^{2}$ be
two BSVIEs as in (43) with drivers $g^{1}$ and $g^{2}$ and terminal condition
$\xi_{t}$ and let $\eta^{1}$ and $\eta^{2}$ be the corresponding families
defined in (54).

a) If $\eta^{1}(r;t,\xi_{t})\leq\eta^{2}(r;t,\xi_{t})$ for any $t\in[0,T]$,
$r\in[t,T]$ and $\xi_{t}\in L^{2}(\mathcal{F}_{T})$, then $g^{1}\leq g^{2}$,
that is $g^{1}(t,v,z)\leq g^{2}(t,v,z)$ for any
$t\in[0,T],v\in[t,T],z\in\mathbb{R}^{d}$.

b) If $Y_{t}^{1,X}\leq Y_{t}^{2,X}$ for any $t\in[0,T]$ and $X\in
L^{2}(\mathcal{F}_{T})$ and if $g^{i}(t,v,0)=0$ for $i=1,2$, then $g^{1}\leq
g^{2}$.

We recall that, for BSDEs, it is sufficient to require that
$\mathcal{E}^{g^{1}}(X)\leq\mathcal{E}^{g^{2}}(X)$ holds for any $X\in
L^{2}(\mathcal{F}_{T})$ to guarantee that
$\mathcal{E}^{g^{1}}_{t}(X)\leq\mathcal{E}^{g^{2}}_{t}(X)$ for any $X\in
L^{2}(\mathcal{F}_{T})$. This is mainly due to time-consistency of
$g$-expectations. See Theorem 4.4 of Briand et al. [8] for details. The proof
of item b) is based on the argument above.

Proof. a) By (54), $\eta^{1}(\cdot;t)$ and $\eta^{2}(\cdot;t)$ are two BSDEs
parameterized by $t$. By applying the Converse Comparison Theorem for BSDEs
(see Theorem 4.1 of Briand et al. [8] and Lemma 2.1 of Jiang [18]) to
$\eta^{1}(\cdot;t),\eta^{2}(\cdot;t)$, it follows that
$\bar{g}^{1}(v,\zeta;t)\leq\bar{g}^{2}(v,\zeta;t)$ for any $v\geq r\geq t$ and
$\zeta\in\mathbb{R}^{d}$. Hence $g^{1}(t,v,z)\leq g^{2}(t,v,z)$ for any
$t\in[0,T],v\in[t,T],z\in\mathbb{R}^{d}$.

b) First of all, we prove that, given two BSDEs with drivers $g^{1}$ and
$g^{2}$ satisfying $g^{i}(v,0)=0$ and a fixed $t$,

$\displaystyle\mathcal{E}^{g^{1}}(X|\mathcal{F}_{t})\leq\mathcal{E}^{g^{2}}(X|\mathcal{F}_{t})\mbox{
for any }X\in L^{2}(\mathcal{F}_{T})$
$\displaystyle\Rightarrow\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\leq\mathcal{E}^{g^{2}}(X|\mathcal{F}_{r})\mbox{
for any }X\in L^{2}(\mathcal{F}_{T})\mbox{ and }r\in[t,T].$ (55)

This implication extends a similar one in the proof of Theorem 4.4 of Briand
et al. [8] where $t=0$. Assume now that, for a given fixed $t$,
$\mathcal{E}^{g^{1}}(X|\mathcal{F}_{t})\leq\mathcal{E}^{g^{2}}(X|\mathcal{F}_{t})$
for any $X\in L^{2}(\mathcal{F}_{T})$. By using a similar approach to those in
Theorem 4.4 of Briand et al. [8] and Lemma 4.5 of Coquet et al. [9],

$\displaystyle
0=\mathcal{E}^{g^{1}}\left(\left.X-\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\right|\mathcal{F}_{t}\right)$
$\displaystyle\leq$
$\displaystyle\mathcal{E}^{g^{2}}\left(\left.X-\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\right|\mathcal{F}_{t}\right)$
$\displaystyle=$
$\displaystyle\mathcal{E}^{g^{2}}\left(\left.\mathcal{E}^{g^{2}}\left(\left.X-\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\right|\mathcal{F}_{r}\right)\right|\mathcal{F}_{t}\right)$
$\displaystyle=$
$\displaystyle\mathcal{E}^{g^{2}}\left(\left.\mathcal{E}^{g^{2}}(X|\mathcal{F}_{r})-\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\right|\mathcal{F}_{t}\right)$

for any $X\in L^{2}(\mathcal{F}_{T})$ and $r\in[t,T]$, where the first and the
last equality are due to translation invariance. Fix now $r\in[t,T]$
arbitrarily and consider $\xi=X1_{A}$ with
$A=\\{\mathcal{E}^{g^{2}}(X|\mathcal{F}_{r})<\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\\}\in\mathcal{F}_{r}$.
On the one hand, by the arguments above,

$\mathcal{E}^{g^{2}}\left(\left.\mathcal{E}^{g^{2}}(\xi|\mathcal{F}_{r})-\mathcal{E}^{g^{1}}(\xi|\mathcal{F}_{r})\right|\mathcal{F}_{t}\right)\geq
0.$

On the other hand,

$\displaystyle\mathcal{E}^{g^{2}}\left(\left.\mathcal{E}^{g^{2}}(\xi|\mathcal{F}_{r})-\mathcal{E}^{g^{1}}(\xi|\mathcal{F}_{r})\right|\mathcal{F}_{t}\right)$
$\displaystyle=$
$\displaystyle\mathcal{E}^{g^{2}}\left(\left.1_{A}\left(\mathcal{E}^{g^{2}}(X|\mathcal{F}_{r})-\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\right)\right|\mathcal{F}_{t}\right)$
$\displaystyle\leq$ $\displaystyle\mathcal{E}^{g^{2}}(0|\mathcal{F}_{t})=0,$

because
$\mathcal{E}^{g^{i}}(X1_{A}|\mathcal{F}_{r})=1_{A}\mathcal{E}^{g^{i}}(X|\mathcal{F}_{r})$
for any $A\in\mathcal{F}_{r}$ holds because of normalization of $g^{i}$ (see
Peng [21]) and the last equality is due to $g^{2}(t,v,0)=0$. Hence,

$\mathcal{E}^{g^{2}}\left(\left.1_{A}\left(\mathcal{E}^{g^{2}}(X|\mathcal{F}_{r})-\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\right)\right|\mathcal{F}_{t}\right)=0.$

Since
$1_{A}\left(\mathcal{E}^{g^{2}}(X|\mathcal{F}_{r})-\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\right)\leq
0$ and strictly negative with probability equal to $P(A)$, strict monotonicity
of conditional $g$-expectation implies that

$1_{A}\left(\mathcal{E}^{g^{2}}(X|\mathcal{F}_{r})-\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})\right)=0,\quad
P-a.s.,$

then
$\mathcal{E}^{g^{2}}(X|\mathcal{F}_{r})\leq\mathcal{E}^{g^{1}}(X|\mathcal{F}_{r})$,
$P$-a.s., for $r\in[t,T]$. This concludes the proof of (55).

Going back to BSVIEs, denote by $\eta^{t,X}_{i,r}=\eta^{i}(r;t,X)$ and by
$\bar{g}^{t}_{i}=\bar{g}^{i}(\cdot,\cdot;t)$ for $i=1,2$, where $\eta$ and
$\bar{g}$ are defined above. Fix now $t$ arbitrarily. Assuming that
$Y_{t}^{1,X}\leq Y_{t}^{2,X}$ for any $X\in L^{2}(\mathcal{F}_{T})$ is
equivalent to assuming that $\eta^{t,X}_{1,t}\leq\eta^{t,X}_{2,t}$ or, also,
to
$\mathcal{E}^{\bar{g}^{t}_{1}}(X|\mathcal{F}_{t})\leq\mathcal{E}^{\bar{g}^{t}_{2}}(X|\mathcal{F}_{t})$
for any $X\in L^{2}(\mathcal{F}_{T})$. By (55), it follows that
$Y_{t}^{1,X}\leq Y_{t}^{2,X}$ for any $X\in L^{2}(\mathcal{F}_{T})$ implies
that
$\mathcal{E}^{\bar{g}^{t}_{1}}(X|\mathcal{F}_{r})\leq\mathcal{E}^{\bar{g}^{t}_{2}}(X|\mathcal{F}_{r})$
for any $r\in[t,T]$, hence $\eta^{t,X}_{1,r}\leq\eta^{t,X}_{2,r}$ for any
$r\in[t,T]$. The thesis then follows by item a) where it is not necessary to
have $\xi_{t}$ but it is enough to consider $X\in L^{2}(\mathcal{F}_{T})$.

####  4.1.2 H-longevity and BSVIEs

The next results investigate under which conditions on the driver $g$
h-longevity of $(\rho_{tu})_{t,u}$ is fulfilled.

###### Corollary 24

If $g(t,\cdot,\cdot)$ is decreasing in $t$ and $g(t,v,0)\geq 0$ for any $t\leq
v$, then h-longevity holds.

Proof. Since $g(t,\cdot,\cdot)$ is decreasing in $t$, sub time-consistency
follows by Theorem 22. We then show here $\rho_{tu}(0)\geq 0$ for any $t\leq
u$, so that we can conclude by Remark 6. We prove now that $g(t,v,0)\geq 0$
for any $t\leq v$ implies that $\rho_{tu}(0)\geq 0$ for any $t\leq u$. Indeed,

$\rho_{tu}(0)=Y_{t}=\int_{t}^{u}g(t,v,Z(t,v))dv-\int_{t}^{u}Z(t,v)dv.$

Since the solution is $(Y_{t}=\int_{t}^{u}g(t,v,0)dv;Z(t,v)=0)$, then
$\rho_{tu}(0)\geq 0$ by the assumption $g(t,v,0)\geq 0$ for any $t\leq v$.

In reality, we can do something more.

###### Proposition 25

If $g(s,v,0)\geq 0$ for any $s\leq v$, then h-longevity holds. Furthermore,
$\gamma(s,t,u,X)=E_{\widetilde{Q}_{s,X}}\left[\int_{t}^{u}g(s,v,0)dv|\mathcal{F}_{s}\right]$
for any $s\leq t\leq u$, where $\widetilde{Q}_{s,X}$ is a suitable probability
measure depending on $X$, with density

$\frac{d\widetilde{Q}_{s,X}}{dP}\triangleq\exp\left\\{-\frac{1}{2}\int_{s}^{u}|\Delta_{z}g(s,v)|^{2}dv+\int_{s}^{u}\Delta_{z}g(s,v)dB_{v}\right\\}.$

Here above $\Delta_{z}g(v)=(\Delta_{z}^{i}g(v))_{i=1,...,d}$ and

$\Delta_{z}^{i}g(v)\triangleq\frac{g(s,v,Z^{u}(s,v))-g(s,v,\bar{Z}^{t}(s,v))}{d\left(Z^{u,i}(s,v)-\bar{Z}^{t,i}(s,v)\right)}1_{\\{Z^{u,i}(s,v)\neq\bar{Z}^{t,i}(s,v)\\}}.$

Proof. Let $s\leq t\leq u$ and let $X\in L^{2}(\mathcal{F}_{t})$ be fixed
arbitrarily. The risk measures $\rho_{st}$ and $\rho_{su}$ satisfy,
respectively, the following BSVIEs:

$\displaystyle\rho_{st}(X)$ $\displaystyle=$
$\displaystyle-X+\int_{s}^{t}g(s,v,Z^{t}(s,v))dv-\int_{s}^{t}Z^{t}(s,v)dB_{v}$
$\displaystyle\rho_{su}(X)$ $\displaystyle=$
$\displaystyle-X+\int_{s}^{u}g(s,v,Z^{u}(s,v))dv-\int_{s}^{u}Z^{u}(s,v)dB_{v}.$

Set now

$\bar{Z}^{t}(s,v)=\left\\{\begin{array}[]{rl}Z^{t}(s,v);&s\leq v\leq t\\\
0;&t<v\leq
u\end{array}\right.;\quad\widetilde{Z}(s,v)=Z^{u}(s,v)-\bar{Z}^{t}(s,v).$

Then

$\displaystyle\rho_{su}(X)-\rho_{st}(X)$ (56) $\displaystyle=$
$\displaystyle\int_{s}^{u}[g(s,v,Z^{u}(s,v))-g(s,v,\bar{Z}^{t}(s,v))]dv+\int_{t}^{u}g(s,v,\bar{Z}^{t}(s,v))dv$
$\displaystyle-\int_{s}^{u}[Z^{u}(s,v)-\bar{Z}^{t}(s,v)]dB_{v}-\int_{t}^{u}\bar{Z}^{t}(s,v)dB_{v}$
$\displaystyle=$
$\displaystyle\int_{s}^{u}[g(s,v,Z^{u}(s,v))-g(s,v,\bar{Z}^{t}(s,v))]dv+\int_{t}^{u}g(s,v,0)dv$
$\displaystyle-\int_{s}^{u}\widetilde{Z}(s,v)dB_{v}$ $\displaystyle=$
$\displaystyle\int_{s}^{u}\Delta_{z}g(s,v)\cdot\widetilde{Z}(s,v)dv-\int_{s}^{u}\widetilde{Z}(s,v)dB_{v}+\int_{t}^{u}g(s,v,0)dv.$

Furthermore, (56) can be rewritten as a linear BSVIE

$\delta\rho_{s}=\Gamma^{t,u}_{s}+\int_{s}^{u}\Delta_{z}g(s,v)\cdot\widetilde{Z}(s,v)dv-\int_{s}^{u}\widetilde{Z}(s,v)dB_{v},$
(57)

where $\delta\rho_{s}\triangleq\rho_{su}(X)-\rho_{st}(X)$ and
$\Gamma^{t,u}_{s}\triangleq\int_{t}^{u}g(s,v,0)dv$ represents the final
condition at time $u$ (which depends on $t$ and also on $s$).

Since $\Gamma^{t,u}\geq 0$ for any $t$ by hypothesis and
$\Delta_{z}g(s,v)\in\mathbb{H}^{2}_{[0,T]}(\mathbb{R}^{d})$ by the assumption
of $g$ Lipschitz in $z$, by the Comparison Theorem on BSVIEs (see Corollary
3.3 of Wang and Yong [28]) it follows the longevity, i.e. $\delta\rho_{s}\geq
0$ for any $s\leq t$. Moreover, by applying Girsanov Theorem, (57) becomes

$\displaystyle\delta\rho_{s}$ $\displaystyle=$
$\displaystyle\int_{t}^{u}g(s,v,0)dv+\int_{s}^{u}\Delta_{z}g(s,v)\cdot\widetilde{Z}(s,v)dv-\int_{s}^{u}\widetilde{Z}(s,v)dB_{v}$
$\displaystyle=$
$\displaystyle\int_{t}^{u}g(s,v,0)dv-\int_{s}^{u}\widetilde{Z}(s,v)dB^{\widetilde{Q}_{s,X}}_{v},$

where $B^{\widetilde{Q}_{s,X}}_{v}\triangleq
B_{v}-B_{s}-\int_{s}^{v}\Delta_{z}g(s,v)dv$, $v\geq s$, is a Brownian motion
with respect to $\widetilde{Q}_{s,X}$ with initial value
$B^{\widetilde{Q}_{s,X}}_{s}=0$. Hence, by taking the conditional expectation
with respect to $\widetilde{Q}_{s,X}$,

$\gamma(s,t,u,X)=\delta\rho_{s}=E_{\widetilde{Q}_{s,X}}\left[\left.\int_{t}^{u}g(s,v,0)dv\right|\mathcal{F}_{s}\right].$

It then follows that
$\rho_{su}(X)-\rho_{st}(X)=E_{\widetilde{Q}_{s,X}}\left[\int_{t}^{u}g(s,v,0)dv|\mathcal{F}_{s}\right]$.

As discussed in Sections 2.2 and 3.1, $\gamma$ may depend on the length of the
time interval $[u,t]$, that is, $\gamma(s,t,u,X)=\gamma_{s,t}(h,X)$ with
$h=u-t$ or, even, $\gamma_{s,t}(h)$ independent from $X$. The following
example provides some cases covering the situation above.

###### Example 26

Let $g(s,v,0)\geq 0$ for any $v\in[s,T]$. Hence, by the result above,
h-longevity holds and
$\gamma(s,t,u,X)=E_{\widetilde{Q}_{s,X}}\left[\int_{t}^{u}g(s,v,0)dv|\mathcal{F}_{s}\right]$
for any $X\in L^{2}(\mathcal{F}_{t})$.

a) If $g(s,v,0)=c_{s}$ for any $v\in[s,T]$, with $c_{s}\geq 0$, then $c_{s}$
is necessarily $\mathcal{F}_{s}$-measurable (since it should be measurable for
any $v\geq s$) and, consequently,

$\gamma(s,t,u,X)=E_{\widetilde{Q}_{s,X}}\left[\left.\int_{t}^{u}g(s,v,0)dv\right|\mathcal{F}_{s}\right]=(u-t)c_{s}.$

In other words, $\gamma$ only depends on the evaluation time $s$ and on
$h=u-t$, that is, roughly speaking, on the length of the time interval over
which there is an uncorrect use of the risk measure ($\rho_{su}$ versus
$\rho_{st}$).

b) If $g(s,v,0)=\exp(-r_{s}\,v)$ for any $v\in[0,T]$, with $r_{s}\geq 0$, then
$r_{s}$ is necessarily $\mathcal{F}_{s}$-measurable (for the same arguments as
above) and, consequently,

$\gamma(s,t,u,X)=\frac{e^{-r_{s}t}\left(1-e^{-r_{s}(u-t)}\right)}{r_{s}}.$

Hence $\gamma$ depends on the evaluation time $s$, on the “right” time horizon
$t$ (referring to the measurability of $X$) and on the length of the time
interval $[t,u]$. Compared to the BSDE case (see Example 12), here $\gamma$
depends also on the evaluation time $s$. This is not surprising for BSVIEs.

Finally, we provide two examples of BSVIEs: the former with a linear driver,
the latter going beyond the Lipschitz case and similarly to Section 3.2.3.

###### Example 27

Consider the driver $g(t,s,z)=a(t,s)\cdot z+b(t,s)$ for any $0\leq t\leq s\leq
T$ and $z\in\mathbb{R}^{d}$, where the $\mathbb{R}^{d}$-valued process
$a(t,s)$ and $1$-dimensional process $b(t,s)$ are given. By applying Girsanov
Theorem, in the same line of Hu and Øksendal [17], the BSVIE associated to the
linear driver above becomes

$\displaystyle Y_{t}$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{T}[a(t,s)\cdot
Z(t,s)+b(t,s)]ds-\int_{t}^{T}Z(t,s)dB_{s}$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{T}b(t,s)ds-\int_{t}^{T}Z(t,s)dB^{\widetilde{Q}_{t}}_{s}$

where
$\frac{d\widetilde{Q}_{t}}{dP}\triangleq\exp\left\\{-\frac{1}{2}\int_{t}^{T}|a(t,s)|^{2}ds+\int_{t}^{T}a(t,s)dB_{s}\right\\}$
and $B^{\widetilde{Q}_{t}}_{u}\triangleq B_{u}-B_{t}-\int_{t}^{u}a(t,s)ds$,
for $u\geq t$, is a Brownian motion with respect to $\widetilde{Q}_{t}$ with
initial value $B^{\widetilde{Q}_{t}}_{t}=0$. Hence, by taking the conditional
expectation with respect to $\widetilde{Q}_{t}$, it holds that

$Y_{t}=E_{\widetilde{Q}_{t}}\left[\left.-X+\int_{t}^{T}b(t,s)ds\right|\mathcal{F}_{t}\right].$

Choosing $b(t,s)\geq 0$, the h-longevity holds.

###### Example 28

Consider the driver $g(t,s,z)=b(t)\frac{|z|^{2}}{2}+a(t,s)$ for any $0\leq
t\leq s\leq T$ and $z\in\mathbb{R}^{d}$, where the deterministic function $b$
is positive and the process $a$ is given. Hence

$\displaystyle Y_{t}$ $\displaystyle=$
$\displaystyle-X+\int_{t}^{T}\left[b(t)\frac{|Z(t,s)|^{2}}{2}+a(t,s)\right]ds-\int_{t}^{T}Z(t,s)dB_{s}$
$\displaystyle=$
$\displaystyle-X+\int_{t}^{T}a(t,s)ds+\int_{t}^{T}b(t)\frac{|Z(t,s)|^{2}}{2}ds-\int_{t}^{T}Z(t,s)dB_{s}$

and, following the same arguments of [27], Example 3.1, it follows that

$Y_{t}=\frac{1}{b(t)}\ln
E_{P}\left[\exp\left\\{-b(t)\left(X-\int_{t}^{T}a(t,s)ds\right)\right\\}\Big{|}\mathcal{F}_{t}\right].$

Whenever $a(t,s)$ is deterministic, $Y_{t}$ becomes

$Y_{t}=\frac{1}{b(t)}\ln
E_{P}\left[e^{-b(t)X}\Big{|}\mathcal{F}_{t}\right]+\int_{t}^{T}a(t,s)ds,$

that is a translation of the usual entropic risk measure. Choosing $a(t,s)>0$,
(strict) h-longevity holds.

###  4.2 Risk measures generated by a family of BSVIEs

Suppose that, for any $t\leq u$, the risk measure $\rho_{tu}$ comes from a
BSVIE with a driver $g_{u}$ depending on the maturity $u$. This means that

$\rho_{tu}^{\mathcal{G}}(X)=\mathcal{E}^{g_{u},V}\left(-X|\mathcal{F}_{t}\right),\quad
X\in L^{2}(\mathcal{F}_{u}),$ (58)

where $\mathcal{E}^{g_{u},V}\left(\xi|\mathcal{F}_{t}\right)$ denotes the
$Y$-component of the solution $(Y_{t},Z(t,s))_{t,s\in[0,T],s\geq t}$ of the
following BSVIE with driver $g_{u}$:

$Y_{t}=\xi+\int_{t}^{u}g_{u}(t,s,Z(t,s))ds-\int_{t}^{u}Z(t,s)dB_{s}.$ (59)

Assume now that $\mathcal{G}=(g_{u})_{u\in[0,T]}$ is a family of drivers
depending on the maturity $u$, independent of $y$, Lipschitz, and convex in
$z$. Each risk measure $\rho^{\mathcal{G}}_{tu}$ is of type (44).

By applying Theorem 21 with a driver $g_{u}$ parameterized by $u$, in a
Brownian setting $\rho_{tu}^{\mathcal{G}}$ can be represented as

$\rho_{tu}(X)=\rho_{tu}^{\mathcal{G}}(X)=\underset{Q_{t}\in\mathcal{Q}_{tu}}{\operatorname{ess.\\!sup}}\left\\{E_{Q_{t}}\left[-X|\mathcal{F}_{t}\right]-\alpha^{\mathcal{G}}_{tu}(Q_{t})\right\\},\quad
X\in L^{2}(\mathcal{F}_{u}),$

where $\mathcal{Q}_{tu}$ is defined in (45), $g_{u}^{*}(t,s,\cdot)$ denotes
the convex conjugate of $g_{u}(t,s,\cdot)$ and the minimal penalty functional
is given by

$\alpha^{\mathcal{G}}_{tu}(Q_{t})=E_{Q_{t}}\left[\left.\int_{t}^{u}g_{u}^{*}(t,s,q(t,s))ds\right|\mathcal{F}_{t}\right].$
(60)

Furthermore, if $g_{u}(t,s,0)=0$ for any $t\leq s\leq u$ then $\rho_{tu}(0)=0$
for any $u$. Differently from the risk measures generated by a single BSVIE
but similarly to those generated by a family of BSDEs, in general
$g_{u}(t,s,0)=0$ for any $t\leq s\leq u$ does not imply the restriction
property.

The following result shows that, for $g_{u}(t,s,0)=0$ for any $t,s,u$, the
restriction property is satisfied only for risk measures induced by a single
BSVIE. This result is not surprising in view of Proposition 13 for the case of
BSDEs with a family of drivers.

###### Proposition 29

Let $g_{u}(t,s,0)=0$ for any $t,s,u$. The restriction property (2) holds if
and only if $g_{u}$ is constant in $u$.

Proof. Assume that the restriction property holds, i.e.
$\rho_{tu}(X)=\rho_{tv}(X)$ for any $t\leq u\leq v$ and $X\in
L^{2}(\mathcal{F}_{u})$. Similarly to (54), denote by

$\eta^{t,X}_{ru}=-X+\int_{r}^{u}\bar{g}_{u}^{t}(s,\zeta_{v}^{t})dv-\int_{r}^{u}\zeta_{v}^{t}dB_{v},\quad
r\in[t;u]$

where

$\zeta^{t}_{v}=Z(t,v);\quad\bar{g}_{u}^{t}(v,\zeta^{t}_{v})=g(t,v,Z(t,v))\quad\mbox{and}\quad\rho_{tu}(X)=\eta^{t,X}_{tu}.$

Assumptions on $g_{u}$ guarantee that $\bar{g}^{t}_{u}(s,0)=0$ for any $s$ and
that $\bar{g}^{t}_{u}(s,\cdot)$ is continuous in $s$. Proceeding as in the
proof of the Converse Comparison Theorem of Briand et al. [8], Thm. 4.1, and
of Jiang [18], Lemma 2.1,

$\displaystyle\bar{g}^{t}_{u}(s,z)=\lim_{\varepsilon\to
0}\frac{\eta^{t,z\cdot(B_{s+\varepsilon}-B_{\varepsilon})}_{su}}{\varepsilon}$
$\displaystyle\bar{g}^{t}_{v}(s,z)=\lim_{\varepsilon\to
0}\frac{\eta^{t,z\cdot(B_{s+\varepsilon}-B_{\varepsilon})}_{sv}}{\varepsilon},$

with convergence in $L^{p}$ with $p\in[1,2)$, for any $z\in\mathbb{R}^{d}$,
$u\leq v$ and $s\in[0,u]$. By extracting a subsequence to obtain convergence
$P$-a.s. and passing to the limit as $\varepsilon\to 0$, it holds that

$\frac{\eta^{t,z\cdot(B_{s+\varepsilon}-B_{\varepsilon})}_{sv}}{\varepsilon}=\frac{\eta^{t,z\cdot(B_{s+\varepsilon}-B_{\varepsilon})}_{su}}{\varepsilon}\longrightarrow\bar{g}^{t}_{u}(s,z),\quad\epsilon\to
0,\>P\mbox{-a.s.}$

where the equality is due to restriction. The thesis then follows because

$\frac{\eta^{t,z\cdot(B_{s+\varepsilon}-B_{\varepsilon})}_{sv}}{\varepsilon}\longrightarrow\bar{g}^{t}_{v}(s,z),\quad\epsilon\to
0,\>P\mbox{-a.s.}$

The converse follows immediately by Proposition 20.

####  4.2.1 Time-consistency

The following result provides a necessary and sufficient condition for a
fully-dynamic risk measure induced by a family of BSVIEs to satisfy sub time-
consistency. Note that the condition on the monotonicity of
$g_{\cdot}(t,\cdot,\cdot)$ is the same as for a BSVIE with a single driver,
while the condition on the monotonicity of the family $g_{u}$ is new.

###### Proposition 30

Let $(\rho_{tu})_{t,u}$ be induced by a BSVIE with a family of drivers
$(g_{u})_{u\in[0,T]}$ as in (58).

a) $(\rho_{tu})_{t,u}$ satisfies sub time-consistency if and only if both the
family $\mathcal{G}$ is increasing and $g_{\cdot}(t,\cdot,\cdot)$ is
decreasing in $t$.

b) $(\rho_{tu})_{t,u}$ satisfies time-consistency if and only if
$\mathcal{G}=\\{g\\}$ and $g_{\cdot}(t,\cdot,\cdot)$ is constant in $t$.

Proof. a) Assume sub time-consistency holds. By (60), the penalty term in the
dual representation of $\rho_{tu}$ is given by

$\alpha^{\mathcal{G}}_{tu}(Q_{t})=E_{Q_{t}}\left[\left.\int_{t}^{u}g_{u}^{*}(t,v,q(t,v))dv\right|\mathcal{F}_{t}\right]$

for any $Q_{t}\in\mathcal{Q}_{tu}$. Let $s,t,u\in[0,T]$ with $s\leq t\leq u$
and let $Q_{s}\in\mathcal{Q}_{st},Q_{t}\in\mathcal{Q}_{tu}$ be fixed
arbitrarily. Set now $\bar{Q}$ the pasting of $Q_{s}$ on $[s,t]$ and of
$Q_{t}$ on $[t,u]$, hence $\bar{Q}\in\mathcal{Q}_{su}$. Denote by $q(s,v)$,
$q(t,v)$ and $\bar{q}(s,v)$ the corresponding processes as in (45). From the
characterization of the penalty term for sub time-consistency in Proposition
7(iii) it follows that

$\displaystyle{\displaystyle
E_{\bar{Q}}\left[\left.\int_{s}^{u}g_{u}^{*}(s,v,\bar{q}(s,v))dv\right|\mathcal{F}_{s}\right]}$
$\displaystyle{\displaystyle\leq
E_{\bar{Q}}\left[\left.\int_{s}^{t}g_{t}^{*}(s,v,\bar{q}(s,v))dv\right|\mathcal{F}_{s}\right]+E_{\bar{Q}}\left[\left.E_{\bar{Q}}\left[\left.\int_{t}^{u}g_{u}^{*}(t,v,\bar{q}(t,v))dv\right|\mathcal{F}_{t}\right]\right|\mathcal{F}_{s}\right]},$

hence

$\displaystyle{\displaystyle 0\leq}$ $\displaystyle{\displaystyle
E_{Q_{s}}\left[\left.\int_{s}^{t}\left[g_{t}^{*}(s,v,\bar{q}(s,v))-g_{u}^{*}(s,v,\bar{q}(s,v))\right]dv\right|\mathcal{F}_{s}\right]}$
(61)
$\displaystyle{\displaystyle+E_{Q_{s}}\left[\left.E_{Q_{t}}\left[\left.\int_{t}^{u}\left[g_{u}^{*}(t,v,\bar{q}(t,v))-g_{u}^{*}(s,v,\bar{q}(s,v))\right]dv\right|\mathcal{F}_{t}\right]\right|\mathcal{F}_{s}\right]}.$

Since (61) should hold for any $s\leq t\leq u$ and any
$Q_{s}\in\mathcal{Q}_{st},Q_{t}\in\mathcal{Q}_{tu}$, it follows that

$\left\\{\begin{array}[]{rl}g_{t}^{*}(s,v,\bar{q})\geq
g_{u}^{*}(s,v,\bar{q}),&\quad\mbox{ for any }s\leq v\leq t\leq u\mbox{ and
}\bar{q}\in\mathbb{R}^{d}\\\ g_{u}^{*}(t,v,\bar{q})\geq
g_{u}^{*}(s,v,\bar{q}),&\quad\mbox{ for any }s\leq t\leq v\leq u\mbox{ and
}\bar{q}\in\mathbb{R}^{d}\end{array}\right.$

Hence, $g_{u}$ is increasing in $u$ and $g_{\cdot}(t,\cdot,\cdot)$ is
decreasing in $t$.

Conversely, sub time-consistency of $(\rho_{tu})_{t,u}$ induced by a family of
BSVIEs can be written in the following notation

$\mathcal{E}^{g_{t},V}(\mathcal{E}^{g_{u},V}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})\leq\mathcal{E}^{g_{u},V}(-X|\mathcal{F}_{s})$
(62)

for any $s\leq t\leq u$ and $X\in L^{2}(\mathcal{F}_{u})$. By (59), the right-
hand and left-hand sides of the previous equation can be rewritten,
respectively, as follows:

$\mathcal{E}^{g_{u},V}(-X|\mathcal{F}_{s})=-X+\int_{s}^{u}g_{u}(s,v,\hat{Z}(s,v))\,dv-\int_{s}^{u}\hat{Z}(s,v)\,dB_{v}$
(63)

and

$\displaystyle\mathcal{E}^{g_{t},V}(\mathcal{E}^{g_{u},V}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})$
(64) $\displaystyle=$
$\displaystyle-X+\int_{t}^{u}g_{u}(t,v,\tilde{Z}(t,v))\,dv-\int_{t}^{u}\tilde{Z}(t,v)\,dB_{v}$
$\displaystyle+\int_{s}^{t}g_{t}(s,v,Z(s,v))\,dv-\int_{s}^{t}Z(s,v)\,dB_{v}$
$\displaystyle=$
$\displaystyle-X+\int_{s}^{u}\left[g_{t}(s,v,Z(s,v))1_{[s,t]}(v)+g_{u}(t,v,\tilde{Z}(t,v))1_{(t,u]}(v)\right]\,dv$
$\displaystyle-\int_{s}^{u}\left[Z(t,v)1_{[s,t]}(v)+\tilde{Z}(t,v)1_{(t,u]}(v)\right]\,dB_{v}.$

Furthermore, (64) becomes

$\displaystyle\mathcal{E}^{g_{t},V}(\mathcal{E}^{g_{u},V}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})$
(65) $\displaystyle\leq$
$\displaystyle-X+\int_{s}^{u}\left[g_{t}(s,v,Z(s,v))1_{[s,t]}(v)+g_{u}(s,v,\tilde{Z}(t,v))1_{(t,u]}(v)\right]\,dv$
$\displaystyle-\int_{s}^{u}\left[Z(t,v)1_{[s,t]}(v)+\tilde{Z}(t,v)1_{(t,u]}(v)\right]\,dB_{v}$
$\displaystyle\leq$
$\displaystyle-X+\int_{s}^{u}\left[g_{u}(s,v,Z(s,v))1_{[s,t]}(v)+g_{u}(s,v,\tilde{Z}(t,v))1_{(t,u]}(v)\right]\,dv$
$\displaystyle-\int_{s}^{u}\left[Z(t,v)1_{[s,t]}(v)+\tilde{Z}(t,v)1_{(t,u]}(v)\right]\,dB_{v},$

where the former inequality is due to decreasing monotonicity of
$g_{\cdot}(t,\cdot,\cdot)$, the latter from increasing monotonicity of the
family of drivers. By setting

$\bar{Z}(s,v)=Z(s,v)1_{[s,t]}(v)+\tilde{Z}(t,v)1_{(t,u]}(v),$

(65) becomes

$\mathcal{E}^{g_{t}}(\mathcal{E}^{g_{u}}(-X|\mathcal{F}_{t})|\mathcal{F}_{s})\leq-X+\int_{s}^{u}g_{u}(s,v,\bar{Z}(s,v))\,dv-\int_{s}^{u}\bar{Z}(s,v)\,dB_{v}.$
(66)

Sub time-consistency then follows by comparing (66) and (63) and by the
uniqueness of the solution of a BSVIE.

b) The case of time-consistency can be obtained by replacing inequalities with
equalities in the proof above.

####  4.2.2 H-longevity and families of BSVIEs

The following result provides sufficient conditions for h-longevity, similarly
to Proposition 18 for BSDEs.

###### Proposition 31

a) If $\mathcal{G}$ is an increasing family of drivers,
$g_{\cdot}(t,\cdot,\cdot)$ is decreasing in $t$, and $\rho_{tu}(0)\geq 0$ for
any $t\leq u$, then $(\rho_{tu})_{t,u}$ satisfies h-longevity.

b) If $\mathcal{G}$ is an increasing family of drivers and $g_{u}\geq 0$ for
any $u$, then h-longevity holds.

Proof. a) follows by Remark 6 and Proposition 30. b) Let $s\leq t\leq u$ and
$X\in L^{2}(\mathcal{F}_{t})$ be fixed arbitrarily. The risk measures
$\rho_{st}$ and $\rho_{su}$ satisfy, respectively, the following BSVIEs:

$\displaystyle\rho_{st}(X)$ $\displaystyle=$
$\displaystyle-X+\int_{s}^{t}g_{t}(s,v,Z^{t}(s,v))dv-\int_{s}^{t}Z^{t}(s,v)dB_{v}$
$\displaystyle\rho_{su}(X)$ $\displaystyle=$
$\displaystyle-X+\int_{s}^{u}g_{u}(s,v,Z^{u}(s,v))dv-\int_{s}^{u}Z^{u}(s,v)dB_{v}.$

Set now

$\bar{Z}^{t,u}(s,v)=\left\\{\begin{array}[]{rl}Z^{t}(s,v);&s\leq v\leq t\\\
0;&t<v\leq
u\end{array}\right.;\quad\widetilde{Z}(s,v)=Z^{u}(s,v)-\bar{Z}^{t,u}(s,v).$

Then

$\displaystyle\delta\rho_{s}$ $\displaystyle=$
$\displaystyle\rho_{su}(X)-\rho_{st}(X)$ (67) $\displaystyle=$
$\displaystyle\int_{s}^{u}[g_{u}(s,v,Z^{u}(s,v))-g_{t}(s,v,\bar{Z}^{t,u}(s,v))]dv+\int_{t}^{u}g_{t}(s,v,0)dv$
$\displaystyle-\int_{s}^{u}[Z^{u}(s,v)-\bar{Z}^{t,u}(s,v)]dB_{v}$
$\displaystyle\geq$
$\displaystyle\int_{s}^{u}[g_{u}(s,v,Z^{u}(s,v))-g_{u}(s,v,\bar{Z}^{t,u}(s,v))]dv-\int_{s}^{u}\widetilde{Z}(s,v)dB_{v}$
$\displaystyle=$
$\displaystyle\int_{s}^{u}\Delta_{z}g_{u}(s,v)\cdot\widetilde{Z}(s,v)dv-\int_{s}^{u}\widetilde{Z}(s,v)dB_{v},$

where the inequality is due to the hypothesis on the drivers and where

$\Delta_{z}^{i}g_{u}(s,v)\triangleq\frac{g_{u}(s,v,Z^{u}(s,v))-g_{u}(s,v,\bar{Z}^{t,u}(s,v))}{d\left(Z^{u,i}(s,v)-\bar{Z}^{t,u,i}(s,v)\right)}1_{\\{Z^{u,i}(s,v)\neq\bar{Z}^{t,u,i}(s,v)\\}}$

for $i=1,...,d$. By Girsanov Theorem, (67) becomes

$\displaystyle\delta\rho_{s}$ $\displaystyle\geq$
$\displaystyle-\int_{s}^{u}\widetilde{Z}(s,v)dB^{\widetilde{Q}_{s}}_{v},$

where
$\frac{d\widetilde{Q}_{s}}{dP}\triangleq\exp\left\\{-\frac{1}{2}\int_{s}^{u}|\Delta_{z}g_{u}(s,v)|^{2}dv+\int_{s}^{u}\Delta_{z}g_{u}(s,v)dB_{v}\right\\}$
and $B^{\widetilde{Q}_{s}}_{v}\triangleq
B_{v}-B_{s}-\int_{s}^{v}\Delta_{z}g_{u}(s,v)dv$, for $v\geq s$, is a Brownian
motion with respect to $\widetilde{Q}_{s}$ with initial value
$B^{\widetilde{Q}_{s}}_{s}=0$. Hence, by taking the conditional expectation
with respect to $\widetilde{Q}_{s}$,

$\delta\rho_{s}\geq
E_{\widetilde{Q}_{s}}\left[\left.-\int_{s}^{u}\widetilde{Z}(s,v)dB^{\widetilde{Q}_{s}}_{v}\right|\mathcal{F}_{s}\right]=0.$

This completes the proof.

In the spirit of Example 19, we can easily extend Example 27 and Example 28 to
cover the case of families of BSVIEs.

Acknowledgements. We thank Tomasz Bielecki, Matteo Burzoni, Igor Cialenco,
Alessandro Doldi, Nicole El Karoui, Mario Ghossoub, Michael Kupper, Felix
Liebrich, Max Nendel, and Frank Riedel for their interest and comments. The
research leading to these results has received funding from the Research
Council of Norway (RCN) within the project STORM - Stochastics for time-space
risk models (nr. 274410). In particular, the second author thanks this
research group for the warm hospitality during her visits. The second author
is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le
loro Applicazioni (GNAMPA), Italy.

## References

  * [1] Acciaio, B., Penner, I. (2011). Dynamic risk measures. In Advanced mathematical methods for finance (pp. 1–34). Springer, Berlin, Heidelberg.
  * [2] Agram, N. (2019). Dynamic risk measure for BSVIE with jumps and semimartingale issues. Stochastic Analysis and Applications, 37(3), 361-376.
  * [3] Barrieu, P., El Karoui, N. (2009). Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Volume on Indifference Pricing (ed: Rene Carmona), Princeton University Press.
  * [4] Bielecki, T. R., Cialenco, I., Pitera, M. (2017). A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective. Probability, Uncertainty and Quantitative Risk 2(1), 3.
  * [5] Bion-Nadal, J. (2008). Dynamic risk-measures: Time consistency and risk measures from BMO martingales. Finance and Stochastics, 12, 219–244.
  * [6] Bion-Nadal, J. (2009). Time consistent dynamic risk processes. Stochastic Processes and their Applications, 119(2), 633–654.
  * [7] Bion-Nadal, J., Di Nunno, G. (2020). Fully-dynamic risk-indifference pricing and no-good-deal bounds. SIAM Journal on Financial Mathematics, 11(2), 620–658.
  * [8] Briand, P., Coquet, F., Hu, Y., Mémin, J., Peng, S. (2000). A converse comparison theorem for BSDEs and related properties of g-expectation. Electronic Communications in Probability, 5, 101–117.
  * [9] Coquet, F., Hu, Y., Mémin, J., Peng, S. (2002). Filtration-consistent nonlinear expectations and related g-expectations. Probability Theory and Related Fields, 123(1), 1-27.
  * [10] Delbaen, F., Peng, S., Rosazza Gianin, E. (2010). Representation of the penalty term of dynamic concave utilities. Finance and Stochastics, 14(3), 449–472.
  * [11] Detlefsen, K., Scandolo, G. (2005). Conditional and dynamic convex risk measures. Finance and Stochastics, 9(4), 539–561.
  * [12] Di Bartolomeo, D. (2017). Risk, Uncertainty and Time Horizon: What Most Risk Models Get Wrong! Newport Seminar, June 2017. Available at: https://www.northinfo.com/documents/770.pdf
  * [13] El Karoui, N., Peng, S., Quenez, M. C. (1997). Backward stochastic differential equations in finance. Mathematical Finance 7(1), 1–71.
  * [14] El Karoui, N., Ravanelli, C. (2009). Cash subadditive risk measures and interest rate ambiguity. Mathematical Finance 19(4), 561–590.
  * [15] Föllmer, H., Schied, A. (2011). Stochastic Finance, third edition, De Gruyter, Berlin.
  * [16] Fuhrmann, S., Kupper, M., Nendel, M. (2022). Wasserstein perturbations of Markovian transition semigroups. arXiv preprint, arXiv:2105.05655v2.
  * [17] Hu, Y., Øksendal, B. (2019). Linear Volterra backward stochastic integral equations. Stochastic Processes and their Applications, 129(2), 626-633.
  * [18] Jiang, L. (2008). Convexity, translation invariance and subadditivity for g-expectations and related risk measures. The Annals of Applied Probability, 18(1), 245-258.
  * [19] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. The Annals of Probability, 28(2), 558-602.
  * [20] Laeven, R. J., Stadje, M. (2014). Robust portfolio choice and indifference valuation. Mathematics of Operations Research, 39(4), 1109-1141.
  * [21] Peng, S. (1997). BSDE and related g-expectations. El Karoui N., Mazliak L. Pitman Research Notes in Mathematics Series, (364), 141–159.
  * [22] Peng, S. (2005). Dynamically consistent nonlinear evaluations and expectations. arXiv preprint math/0501415.
  * [23] Quenez, M. C., Sulem, A. (2013). BSDEs with jumps, optimization and applications to dynamic risk measures. Stochastic Processes and their Applications, 123(8), 3328-3357.
  * [24] Revuz, D., Yor, M. (2013). Continuous martingales and Brownian motion (Vol. 293). Springer Science & Business Media.
  * [25] Rosazza Gianin, E. (2006). Risk measures via g-expectations. Insurance: Mathematics and Economics, 39(1), 19-34.
  * [26] Royer, M. (2006). Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic processes and their applications, 116(10), 1358-1376.
  * [27] Wang, H., Sun, J., Yong, J. (2021). Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations. Applied Mathematics & Optimization, 84(1), 145-190.
  * [28] Wang, T., Yong, J. (2015). Comparison theorems for some backward stochastic Volterra integral equations. Stochastic Processes and their Applications, 125(5), 1756-1798.
  * [29] Yong, J. (2007). Continuous-time dynamic risk measures by backward stochastic Volterra integral equations. Applicable Analysis, 86(11), 1429-1442.
  * [30] Yong, J. M. (2013). Backward stochastic Volterra integral equations - a brief survey. Applied Mathematics-A Journal of Chinese Universities, 28(4), 383-394.