# An ergodic BSDE approach to forward entropic risk measures: representation and large-maturity behavior

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## Abstract

Using elements from the theory of ergodic backward stochastic differential equations (BSDEs), we study the behavior of forward entropic risk measures in stochastic factor models. We derive general representation results (via both BSDEs and convex duality) and examine their asymptotic behavior for risk positions of large maturities. We also compare them with their classical counterparts and provide a parity result.

## Keywords

Forward entropic risk measures Stochastic factor models Ergodic BSDE Convex duality representation Large-maturity behavior## Mathematics Subject Classification (2010)

91G10 91G80 60H30## JEL Classification

G11 D81## 1 Introduction

Risk measures constitute one of the most active areas of research in financial mathematics, for they provide a general axiomatic framework to assess risks. Their universality and wide applicability, together with the wealth of related interesting mathematical questions, have led to considerable theoretical and applied developments; see, among others, [1, 11] and [10, Chap. 4] with more references therein, and [3, 6, 21, 34] for dynamic convex risk measures.

A number of popular risk measures are defined in relation to investment opportunities in a financial market like, for example, VaR, CVaR, indifference prices, etc. However, such measures are directly tied to a specific trading horizon, for it is implicitly assumed that every risk position will be introduced and mature at times (random or not) up to this pre-chosen horizon. We refer the reader to [36, Sect. 3.3] for motivational examples and detailed discussions. Such issues motivated the authors of [36] to introduce the so-called *maturity-independent risk measures* (Definition 3.1 therein). These measures were defined axiomatically, in relation to a general semimartingale market, via four fundamental properties: *anti-positivity, convexity, cash translativity* and *replication invariance*.

An important class of maturity-independent risk measures are the *forward entropic risk measures*, which were constructed as (negative) indifference prices under exponential forward performance criteria (see Definition 3.1). Forward performance criteria are quite appropriate to define dynamic risk measures that do not depend on a specific horizon or maturity, since the underlying forward performance processes are defined for all times. As a result, they can be suitably used to assess the performance of investment strategies with and without the (arbitrary) risk positions, no matter when these positions are introduced or mature. For general semimartingale models, an implicit form for forward entropic risk measures was derived in [36] (cf. (4.14) therein) via the solution of a forward stochastic optimization problem.

Herein, we build on the work of [36], focusing on incomplete market models with multiple stocks and multiple stochastic factors, and working with forward performance processes that are deterministic functions of these factors. This is a rich class of criteria that not only offer tractability in constructing and further studying the forward risk measures, but also enable us to make interesting connections with ergodic backward stochastic differential equations (BSDEs).

Markovian forward performance criteria in stochastic factor models were studied in [32], where the multi-stock/multi-factor complete market case was solved. The incomplete market case with a single stock/single factor was examined in [33] and in [35] for a model with slow and fast stochastic factors, and more recently in [23].

While we look at a smaller class of market models than those considered in [36], our setting is more general in two directions. Firstly, we incorporate trading constraints. This not only makes the analysis more involved but, more importantly, results in violation of the replication invariance property (see discussions after Corollary 3.6). Secondly, we consider and analyze in detail the forward risk measure process, while the authors of [36] only provided the risk measure at a single time, through an implicit forward optimization problem.

Our contribution is multifold. Firstly, we study the explicit representation of the forward entropic risk measure process in Theorem 3.2. To obtain it, we need to solve two forward stochastic optimization problems, with and without the risk position for an arbitrary trading horizon associated with the maturity of the (arbitrary) risk position. The solution of the latter problem is given directly by the exponential forward performance process itself, but it is not a priori clear how to solve the former, even if we interpret it as a classical expected utility problem with random endowment. This is due to the presence of additional terms that violate standard boundedness or integrability conditions. As a consequence, classical results in expected utility theory do not apply (see discussions after Definition 3.1). Instead, we use an alternative approach, based on the recent work of [23] on the construction of homothetic (exponential, power and logarithmic) forward performance processes using ergodic BSDEs. This method bypasses a number of technical difficulties associated with solving an underlying ill-posed stochastic partial differential equation (SPDE) that the forward performance process is expected to satisfy; see [7] and [31] for discussions on this forward SPDE.

For the exponential family we consider herein, the approach in [23] yields a unique representation of the forward performance criterion in a factor form (Proposition 2.6). Based on this result, we establish in Theorem 3.2 that the forward entropic risk measure satisfies a BSDE whose driver depends on the solution of the ergodic BSDE for the forward performance process.

From this BSDE representation, we establish the following results. Firstly, using the convexity property of its driver, we derive a convex dual representation of the forward entropic risk measure (Theorem 3.5). Specifically, we show that the forward entropic risk measure is the minimal, among all equivalent probability measures, expected value of the risk position plus a penalty term. The penalty term has the following properties: it is independent of both the risk position and its maturity. Rather, it depends exclusively on the stochastic factors and the solution to the aforementioned ergodic BSDE.

The dual representation result readily yields that the three properties—*anti-positivity*, *convexity* and *cash translativity*—that were introduced in [36] indeed hold. It also demonstrates that because of trading constraints, the *replication invariance* property fails. If the constraints are removed, this property naturally holds; this can also be seen in the example in Sect. 4, where the equivalent probability measures turn out to be equivalent martingale measures.

In a different direction, we derive a parity result between the forward and classical entropic risk measures. We show that the former can be constructed as the difference of two classical entropic risk measures applied, respectively, to a modified risk position and a normalizing factor related to the solution of the ergodic BSDE for the forward performance process.

We also study the asymptotic behavior of the forward entropic risk measures when their maturity is long (Theorem 3.10). For risk positions given by deterministic functions of the stochastic factor processes, we show that their risk measures converge to a constant which is independent of the initial state of the stochastic factors, and furthermore, the convergence is exponentially fast. We also derive an explicit exponential bound of the associated hedging strategies, which in turn yields that as the maturity goes to infinity, no trading occurs in any finite time to hedge the underlying risks.

We conclude with an example cast in the single stock/single stochastic factor case. Using the ergodic BSDE approach, we derive a closed form representation of the forward entropic risk measure and its convex dual representation. We also derive a representation of its classical analogue, and in turn compute numerically the long-term limits of the two measures for specific risk positions.

The paper is organized as follows. In Sect. 2, we introduce the stochastic factor market model and provide background results on exponential forward performance processes. In Sect. 3, we provide the definition of the forward entropic risk measures, derive their representation results and also establish the parity with their classical counterparts. We also study the long-maturity behavior. We present an example in Sect. 4 and conclude in Sect. 6. For the reader’s convenience, the proofs of the main results are presented separately in Sect. 5.

## 2 The investment model and the performance criterion

Let \(W\) be a \(d\)-dimensional Brownian motion on a probability space \((\Omega ,\mathcal{F},\mathbb{P})\). Denote by \(\mathbb{F}=( \mathcal {F}_{t})_{t\geq 0}\) the augmented filtration generated by \(W\). We consider a market with a risk-free bond offering zero interest rate and \(n\) risky stocks, with \(n\leq d\).

We introduce the following model assumptions. Throughout, we use \(A^{\mathrm{tr}}\) for the transpose of a matrix \(A\), and whenever needed, the self-evident notation \(V^{v}\) for the stochastic factor process \(V\) starting from \(V_{0}=v\).

### Assumption 2.1

(i) The drift and volatility coefficients \(b^{i}(v) \) and \(\sigma^{i}(v)\), \(v\in\mathbb{R}^{d}\), are uniformly bounded.

(ii) The volatility matrix \(\sigma(v):=(\sigma^{1}(v),\ldots ,\sigma^{n}(v))^{\mathrm{tr}}\), \(v\in\mathbb{R}^{d}\), has full row rank \(n\).

### Assumption 2.2

(ii) The matrix \(\kappa\) is positive definite and normalized to \(\vert\kappa\vert=1\).

The “large enough” property of \(C_{\eta}\) in Assumption 2.2(i) will be quantified in the sequel when it is assumed that \(C_{\eta }>C_{v}>0\), where \(C_{v}\) appears in the properties of the driver of an upcoming ergodic BSDE (see inequality (5.1)).

The assumption that the matrix \(\kappa\) is constant is without loss of generality as long as the more general case \(\kappa=\kappa(v) \) satisfies the uniform ellipticity condition. This condition is crucial for establishing the coupling estimate (2.5) that will be frequently used. The case where \(\kappa (v) \) is degenerate is far more challenging and left for future research.

### Proposition 2.3

*Under Assumption *2.2, *the following assertions hold*:

*The stochastic factor process satisfies*,

*for*\(v,\bar{v}\in \mathbb{R} ^{d}\), \(t\geq0\),

*Assume that the process*\(V^{v}\)

*follows*

*where*\(H:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\)

*is a bounded measurable function*, \(\mathbb{Q}^{H}\)

*is a probability measure equivalent to*ℙ,

*and*\(W^{H}\)

*is a*\(\mathbb{Q}^{H}\)-

*Brownian motion*.

*Then there exists a constant*\(C>0\)

*such that*

*for any*\(p\geq1\).

*Furthermore*,

*for any measurable function*\(\phi: \mathbb{R}^{d}\rightarrow\mathbb{R}\)

*with polynomial growth rate*\(\mu>0\)

*and*\(v,\bar{v}\in\mathbb{R}^{d}\),

*The constants*\(C\)

*and*\(\hat{C}_{\eta}\)

*depend on the function*\(H(\cdot)\)

*only through*\(\sup_{v\in\mathbb{R}^{d}}|H(v)|\).

The proof of (i) follows from Gronwall’s inequality. Inequality (2.4) is an application of a Lyapunov argument (see [9, Lemma 3.1]), while inequality (2.5) follows from the coupling estimate in [19, Lemma 3.4].

Proposition 2.3 implies that the stochastic factor process \(V\) admits a unique invariant measure and is thus ergodic. Moreover, any two paths converge to each other exponentially fast.

### 2.1 The trading strategies

*all*\(t\geq0\).

### 2.2 The forward performance criterion

The investor uses an exponential forward performance criterion to measure the performance for her investment strategies. For the reader’s convenience, we start with some background results about this criterion, first recalling its definition (see [26, 27, 28, 29, 30]). We then focus on the exponential class and review its ergodic BSDE representation established in [23].

### Definition 2.4

Let \(\mathbb{D=R}\times[ 0,\infty)\). A process \(U(x,t) \), \((x,t) \in \mathbb{D}\), is a *forward performance process* if

(i) for each \(x\in\mathbb{R}\), \(U(x,t) \) is \(\mathbb{F}\)-progressively measurable;

(ii) for each \(t\geq0\), the mapping \(x\mapsto U(x,t)\) is strictly increasing and strictly concave;

*Markovian exponential forward performance criteria,*namely, processes that are

*deterministic functions*of the stochastic factors,

There are two approaches in specifying the process \(f( V_{t},t)\). One may try to determine the function(s) \(f( \cdot,\cdot) \) using a (deterministic) PDE that they are expected to satisfy. The form of this equation follows from imposing on the candidate solutions (2.7) the supermartingale and martingale conditions in part (iii) above (see [23]; also [31] and [32]). However, this PDE is ill-posed, and as a consequence, there are various difficulties and open questions for its solution(s).

Alternatively, one may specify the process \(f( V_{t},t) \) directly. This was done in [23], where the authors developed a probabilistic approach based on ergodic BSDEs (see [4, 5, 12, 19, 23] for recent developments of ergodic BSDEs). They showed that the forward performance process of the form (2.7) exists and furthermore derived its explicit representation using the solution of an ergodic BSDE. For the reader’s convenience, we briefly review these results next.

### Proposition 2.5

([23, Proposition 4.1])

*Suppose that Assumptions*2.1

*and*2.2

*hold*.

*Then the ergodic BSDE*

*where the driver*\(F:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow \mathbb{R} \)

*is defined as*

*with*\(\theta(\cdot)\)

*as in*(2.2),

*admits a unique Markovian solution*\((Y_{t},Z_{t},\lambda)\), \(t\geq0\).

*Specifically*,

*there exist a unique*\(\lambda\in\mathbb{R}\)

*and functions*\(y:\mathbb{R}^{d}\rightarrow\mathbb{R}\)

*and*\(z:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\)

*such that*

*The function*\(y(\cdot)\)

*is unique up to an additive constant*,

*and without loss of generality*,

*we can set*\(y(0)=0\).

*Moreover*, \(y(\cdot)\)

*has at most linear growth and satisfies*

*and*\(z(\cdot)\)

*is bounded with*\(|z(\cdot)|\leq\frac{C_{v}}{C_{\eta }-C_{v}}\),

*where the constants*\(C_{\eta}\)

*and*\(C_{v}\)

*are as in Assumption*2.2

*and inequality*(5.1),

*respectively*.

As shown in [23], the form of the driver \(F\) is dictated by the martingale and supermartingale requirements (see part (iii) in Definition 2.4) that the candidate forward performance process must satisfy. In Sect. 4, we provide a specific example for this driver. We also refer the reader to [23, Sect. 3.1.2] for the connection of \(F\) with the ill-posed PDE that the function \(f(\cdot,\cdot)\) in (2.7) solves.

The next result relates the above unique *Markovian solution* of the ergodic BSDE (2.8) to the *Markovian exponential forward performance process* (2.7) and its associated optimal policy. For its proof, see [23, Theorem 4.2].

### Proposition 2.6

([23, Theorem 4.2])

*Suppose that Assumptions*2.1

*and*2.2

*hold*,

*and let*\((Y,Z,\lambda)\)

*be the unique Markovian solution to the ergodic BSDE*(2.8).

*Then the process*\(U(x,t)\), \((x,t) \in\mathbb{D}\),

*given by*

*is an exponential forward performance process*.

*It solves*

*Furthermore*,

*the associated optimal strategy is given by*

From (2.13), we may identify the process \(Z\) with the *volatility* component of the forward performance criterion. We also remark that the volatility term in the Itô decomposition of \(U(x,t) \) is not zero, and thus \(U(x,t) \) is not monotone with respect to time.

### Remark 2.7

In [23], the solution pair \((Y,Z)\) is constructed by a “vanishing discount rate” argument, i.e., the components of \((Y,Z)\) are the limiting processes, as \(\rho\downarrow0\), of the solution to an infinite-horizon BSDE with a discount factor \(\rho\). Furthermore, the constant \(\lambda\) can be interpreted as the long term growth rate of a classical exponential utility maximization problem (see [23, Proposition 3.3]).

### Remark 2.8

## 3 Forward entropic risk measures

### Definition 3.1

Consider the Markovian forward exponential performance process \(U(x,t) =-e^{-\gamma x+}{}^{Y_{t}-\lambda t}\), \((x,t)\in\mathbb{D}\) (cf. (2.12)).

*-normalized forward entropic risk measure,*denoted by \(\rho_{t}(\xi;T)\in\mathcal {F}_{t}\), is defined for \(t\in[ 0,T] \) via

*forward entropic risk measure,*denoted by \(\rho_{t}(\xi)\), is defined, for \(t\in[ 0,T] \), as

Definition (3.3) highlights the independence of the risk measure operator \(\rho_{t}(\cdot)\) on both the maturity and the trading horizon. Obviously, for each risk position \(\xi\), \(\rho_{t}(\xi)\) depends on the size of \(\xi\) and \(T\). But the operator per se is constructed through (3.1) and (3.2), which are defined for any maturity \(T\) since the underlying forward performance criterion is defined for all times \(t\geq0\).

### 3.1 BSDE representation of forward entropic risk measures

We provide here the first main result, which is the representation of the forward entropic risk measure process using finite horizon BSDEs and ergodic BSDEs. We show that the risk measure process of an arbitrary maturity, say \(T>0\), can be constructed as the solution of a BSDE on \([ 0,T]\), with its driver depending on the process \(Z\) of the solution to the ergodic BSDE (2.8). This dependence emerges because the ergodic BSDE (2.8) was used to construct the forward performance criterion (2.12) appearing in (3.1) of Definition 3.1.

### Theorem 3.2

*Consider an arbitrary risk position*\(\xi\in\mathcal{L}\)

*with maturity*\(T\).

*Suppose that Assumptions*2.1

*and*2.2

*hold*.

*Introduce for*\(t\in[ 0,T] \)

*the BSDE*

*with the driver*\(G:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R} ^{d}\rightarrow\mathbb{R}\)

*defined as*

*where*\(F\)

*is as in*(2.9)

*and the process*\(Z\)

*is the second component of the solution*(2.10)

*of the ergodic BSDE*(2.8).

*Then the following assertions hold*:

i) *The BSDE* (3.4) *has a unique solution*\(( Y_{t}^{\xi},Z_{t}^{\xi}) \), \(t\in[ 0,T] \), *with*\(Y^{\xi } \)*being uniformly bounded and*\(Z^{\xi}\in\mathcal{L}_{\mathrm{BMO}}^{2}[0,T]\).

*The forward entropic risk measure of*\(\xi\)

*is given*,

*for*\(t\in[ 0,T] \),

*by*

An immediate consequence of the above representation is the dynamic consistency property of the forward entropic risk measures.

### Corollary 3.3

The above property follows directly from (3.6) and the fact that the solution of (3.4) satisfies \(Y_{t}^{-\xi}=Y_{t}^{Y_{s}^{-\xi}}=Y_{t}^{\rho_{s}( \xi ) }\), for \(0\leq t\leq s\leq T\).

The fact that the forward entropic risk measure is represented as the solution of a finite horizon BSDE is not surprising since given an arbitrary maturity, the “indifference-type” condition (3.1) is by nature set “backwards” in time. There are, however, two fundamental differences between the BSDEs for classical and forward entropic risk measures. Firstly, the BSDE (3.4) differs from its classical counterpart since its driver \(G\) depends on the process \(Z\) related to the solution of the ergodic BSDE. Thus the two BSDEs do not coincide, and as a result, they produce different solutions within the common horizon \([ 0,T] \). This is, for example, reflected below in the parity result (3.13) between the forward and classical entropic risk measures. Secondly, the BSDE for the classical entropic risk measure is defined only for a preset investment horizon \([ 0,T] \), for some single, fixed \(T\). In contrast, in the forward setting, the BSDE (3.4) is set for any arbitrary interval \([ 0,T] \), associated with the maturity \(T\) of the arbitrary risk position \(\xi\).

Besides defining the forward entropic risk measure of a risk position, one may also introduce the associated forward *hedging strategies*. As in the classical case, they are defined as the difference between the optimal strategies with and without the risk position. The former is the optimal strategy for (3.2), provided in the sequel (see (5.5)), while the latter was derived in (2.14).

### Corollary 3.4

*Consider an arbitrary risk position*\(\xi \in\mathcal{L}\)

*with maturity*\(T\).

*Suppose that Assumptions*2.1

*and*2.2

*hold*.

*Then the associated hedging strategy*\(\alpha_{t,T}\)

*for*\(t\in [0,T]\)

*is given by*

Observe that the hedging strategy naturally depends on the maturity of the risk position in consideration only through the first term \({\pi}_{t}^{\ast,{\xi}}\), and in particular through the process \(Z^{-{\xi}}\). The second term, the optimal policy of \(U( x,t) \), \(t\geq0\), is independent of both the claim and its maturity. This is not the case in the classical setting, where both terms depend on the investment horizon of the underlying exponential utility maximization problems (see for instance [16]).

### 3.2 Convex dual representation of forward entropic risk measures

### Theorem 3.5

*Consider an arbitrary risk position*\(\xi\in \mathcal{L}\)*with maturity*\(T\). *Suppose that Assumptions *2.1*and*2.2*hold*. *Then for*\(t\in[ 0,T] \), *the following assertions hold*:

*The forward entropic risk measure*\(\rho_{t}(\xi)\)

*admits the convex dual representation*

*where*\({\hat{G}}:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R} ^{d}\rightarrow\mathbb{R}\cup\{ \infty\}\),

*the convex dual of the driver*\(G \),

*is defined in*(3.7),

*and the process*\(Z\)

*is the second component in the solution of the ergodic BSDE*(2.8).

*There exists an optimal density process*\(q^{\ast,\xi}\in \mathcal{A} _{[t,T]}^{\prime}\),

*and thus*

In the above representation formula, the convex dual \(\hat{G}\) yields the “penalty” process \({\hat{G}}(V_{s},Z_{s},\cdot)\), \(s\in[0,T]\), which is added to the original risk position. However, it is per se common for *all* claims and independent of *all* maturities.

From the representation (3.11) and the properties of \(\hat {G}\), we have the following result.

### Corollary 3.6

*Consider an arbitrary risk position*\(\xi\in\mathcal{L}\)*with maturity*\(T\). *Suppose that Assumptions *2.1*and *2.2*hold*. *Then for*\(t\in[ 0,T] \), *the following properties hold*:

(i) (*Anti*-*positivity*) \(\rho_{t}( \xi) \leq0\)*for*\(\xi\geq0 \).

(ii) (*Convexity*) *For any*\(\bar{\xi}\in\mathcal{L}^{\infty }(\mathcal{F}_{T})\)*and*\(\alpha\in\mathcal{L}^{\infty}(\mathcal{F}_{t}) \)*with*\(\alpha\in[0,1] \), *we have*\(\rho_{t}( \alpha\xi+(1-\alpha ) \bar{\xi})\leq\alpha\rho_{t}(\xi)+(1-\alpha)\rho_{t}(\bar{\xi}) \).

(iii) (*Cash translativity*) *For any*\(m\in\mathcal{L}^{\infty }(\mathcal{F}_{t}) \), *we have*\(\rho_{t}( \xi -m) =\rho_{t}( \xi) +m\).

*replication invariance*property in general fails, i.e.,

### 3.3 A parity result between forward and classical entropic risk measures

### Definition 3.7

*classical entropic risk measure,*denoted by \(\rho_{t,T}(\xi)\in\mathcal{F}_{t}\), is defined for \(t\in[ 0,T] \) via

The decomposition formula below shows that the forward entropic measure can be constructed as the difference of two classical entropic measures applied, respectively, to the modified risk position \(\xi-\frac{Y_{T}-\lambda T}{\gamma} \) and to a normalizing factor \(-\frac{Y_{T}-\lambda T}{\gamma}\).

### Proposition 3.8

*Consider an arbitrary risk position*\(\xi\in \mathcal{L}\)

*with maturity*\(T\).

*Suppose that Assumptions*2.1

*and*2.2

*hold*.

*Then for*\(t\in[ 0,T] \),

*the forward and classical entropic risk measures satisfy*

*where*\(( Y,\lambda) \)

*is the unique Markovian solution to the ergodic BSDE*(2.8) (

*cf*.

*Proposition*2.5).

### 3.4 Risk positions with long maturity

### Assumption 3.9

The function \(g:{\mathbb{R}^{d}}\rightarrow {\mathbb{R}}\) is uniformly bounded and Lipschitz-continuous, with Lipschitz constant \(C_{g}\).

We establish that as \(T\uparrow\infty\), \(\rho_{0}( \xi ) \) converges to a constant which is independent of the initial value of the stochastic factor and moreover, we prove the rate of convergence. We also study the limiting behavior of the hedging strategies and establish appropriate exponential bounds. These yield that as \(T\uparrow\infty\), the optimal strategies with and without the risk position coincide, and thus at all times, no additional stock trading to hedge the risk position takes place.

### Theorem 3.10

*Consider an arbitrary risk position*\(\xi \)*as in* (3.14) *with maturity*\(T\), *and the function*\(y^{T,g}( \cdot,\cdot) \)*as in* (3.15). *Suppose that Assumptions *2.1, 2.2*and *3.9*hold*. *Then the following assertions hold*:

*For any*\(v\in\mathbb{R}^{d}\)

*and*\(V_{0}=v\),

*there exists a constant*\(L^{g}\in\mathbb{R}\),

*independent of*\(v\),

*such that*

*In particular*,

*for any*\(T>0\),

*for some constant*\(C\),

*where the constant*\(\hat{C}_{\eta}\)

*is given in Proposition*2.3.

*The hedging strategy satisfies for any*\(T>0\)

*and*\(s\in [0,T)\)

*that*

*Therefore*,

*for any*\(s\geq0\),

### Remark 3.11

However, the driver of the ergodic BSDE (3.19) depends on the solution \(Z\) of the ergodic BSDE (2.8) of the forward performance process, and this causes various technical issues. Indeed, the driver \(G(v,z(v),\bar{z})\) of the ergodic BSDE (3.19) *depends* on the function \(z(\cdot)\) in (2.10). Although, due to the boundedness of the function \(z(\cdot)\), the driver \(G\) satisfies the local Lipschitz estimate (5.2) in \(\bar{z}\), it need not satisfy the local Lipschitz estimate (5.1) in \(v\), and hence the existence and uniqueness result in [23] might not apply. Moreover, it is not even clear whether the ergodic BSDE (3.19) is well-posed. For these reasons, we establish the above results working directly with the function \(y^{T,g}(\cdot,\cdot)\) in (3.15).

## 4 An example

We present an example for which we derive explicit formulae for both the forward and classical entropic risk measures, and also provide numerical results for their large-maturity limits.

### 4.1 Forward entropic risk measure

*replication invariance*property, namely,

### 4.2 Classical entropic risk measure

*independent*of the maturity \(T\). In the classical setting, however, the measure \(\mathbb{Q}^{T}\) is determined by the component \(Q_{2}^{0}\) coming from the exponential utility maximization (3.12) with zero risk position (cf. (4.6) with \({\xi}=0\)), which depends critically on the maturity \(T\). Finally, (4.4) and (4.8) yield

### 4.3 Numerical results

Note that \(\kappa_{1}\) is the correlation between the stochastic factor \(V\) and the stock \(S\). The larger \(\kappa_{1}\), the more likely the investor is able to hedge the underlying risks via trading the stock. The numerical results in Figs. 1 and 3 show that in both extreme cases, \(\kappa_{1}\approx1\) or \(\kappa_{1}\approx0\), both the forward and classical entropic risk measures converge to constants more quickly, in comparison to the intermediate case \(\kappa=0.5\). This implies that in these two extreme situations, the investor will implement earlier the “no trading in the stock” strategy.

We should, however, like to mention that it is not yet clear how the two limiting constants compare to each other. This open question is left for future research.

## 5 Proofs of the main results

### 5.1 Proof of Theorem 3.2

### Proof of (i)

### Proof of (ii)

### 5.2 Proof of Theorem 3.5

We first derive some auxiliary bounds for the driver \(G\) and its convex dual \(\hat{G}\).

### Lemma 5.1

*The driver*\(G(v,z,\bar{z})\)*and its convex dual*\({\hat{G}}(v,z,p)\) (*cf*. (3.5) *and* (3.7), *respectively*) *have the following properties*:

*For*\((v,z,\bar{z})\in\mathbb{R}^{d}\times\mathbb {R}^{d}\times\mathbb{R} ^{d}\),

(ii) *For*\((v,z)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\), \({\hat{G}} (v,z,p)\)*is convex in*\(p\).

*For*\((v,z,p)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\times \mathbb{R} ^{d}\),

### Proof

The convexity of \(\hat{G}(v,z,p)\) in \(p\) is immediate, so we only prove (i) and (iii).

### Proof of Theorem 3.5

### 5.3 Proof of Proposition 3.8

### 5.4 Proof of Theorem 3.10

We first establish some auxiliary estimates for the function \(y^{T,g}: \mathbb{R}^{d}\times[0,T]\rightarrow\mathbb{R}\) appearing in (3.15).

### Lemma 5.2

*Consider an arbitrary risk position*\(\xi\)*as in* (3.14) *with maturity*\(T\)*and the function*\(y^{T,g}( \cdot,\cdot) \)*as in* (3.15). *Suppose that Assumptions *2.1, 2.2*and *3.9*hold*. *Then for*\((v,t)\in \mathbb{R}^{d}\times[0,T]\), *the function*\(y^{T,g}(\cdot,\cdot)\)*has the following properties*:

*There exists a constant*\(C>0\)

*such that*

*With the constants*\(K\)

*as in*(A.3), \(C_{v}\)

*in*(5.1)

*and*\(C_{\eta}\)

*in Assumption*2.2(i),

*There exists a constant*\(C>0\)

*such that for*\(v,\bar{v}\in \mathbb{R} ^{d}\),

*with the constant*\(\hat{C}_{\eta}\)

*given in Proposition*2.3.

### Proof

### Proof of Theorem 3.10

(ii) We only establish the exponential bound (3.17), since the asymptotic behavior of \(\alpha _{t,T}\) in (3.18) will then follow by letting \(T\uparrow\infty\).

## 6 Conclusions and extensions

We have studied forward entropic risk measures for stochastic factor models and in the presence of trading constraints. Using the ergodic BSDE representation of the involved exponential forward performance processes, we have established two representation results, working with the primal and the dual domains, respectively. We have also derived a parity result between the forward entropic risk measures and their classical counterparts, and moreover investigated their asymptotic behavior for large maturities.

The approach and the results herein may be extended in several directions. Firstly, one may allow stochastically evolving set of constraints. This is undoubtedly a very important extension, for trading constraints in many applications are affected by upcoming (and frequently, non-anticipated) market changes, past performance and other features that the forward performance criteria may accommodate. To the best of our knowledge, this generalization has not been considered so far in the context of forward performance criteria.

Another important issue is the relative valuation and risk management of incoming projects. Herein, we consider the measurement of risk positions with arbitrary maturities, but in isolation from each other. In various applications, however, one needs to price incoming projects in relation to existing ones, and work with relative risk assessment. In order to do this, one first needs to define properly “relative” forward performance processes, which will naturally depend on the evolving risks associated with the existing projects. Such extensions are left for future research.

## Notes

### Acknowledgements

We thank the Editor, Associate Editor and two anonymous referees for their valuable comments and suggestions. This work was presented at the SIAM Conference on Financial Mathematics and Engineering, Austin, the 9th World Congress of the Bachelier Finance Society, New York, the 9th European Summer School in Financial Mathematics, Pushkin, and the 5th Berlin Workshop on Mathematical Finance for Young Researchers, Berlin. The authors thank the participants for fruitful comments.

## References

- 1.Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance
**9**, 203–228 (1999) MathSciNetCrossRefzbMATHGoogle Scholar - 2.Becherer, D.: Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insur. Math. Econ.
**33**, 1–28 (2003) MathSciNetCrossRefzbMATHGoogle Scholar - 3.Bion-Nadal, J.: Dynamic risk measures: time consistency and risk measures from BMO martingales. Finance Stoch.
**12**, 219–244 (2008) MathSciNetCrossRefzbMATHGoogle Scholar - 4.Cosso, A., Fuhrman, M., Pham, H.: Long time asymptotics for fully nonlinear Bellman equations: a backward SDE approach. Stoch. Process. Appl.
**126**, 1932–1973 (2016) MathSciNetCrossRefzbMATHGoogle Scholar - 5.Debussche, A., Hu, Y., Tessitore, G.: Ergodic BSDEs under weak dissipative assumptions. Stoch. Process. Appl.
**121**, 407–426 (2011) MathSciNetCrossRefzbMATHGoogle Scholar - 6.Detlefsen, K., Scandolo, G.: Conditional and dynamic convex risk measures. Finance Stoch.
**9**, 539–561 (2005) MathSciNetCrossRefzbMATHGoogle Scholar - 7.El Karoui, N., Mrad, M.: An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE. SIAM J. Financ. Math.
**4**, 697–736 (2014) MathSciNetzbMATHGoogle Scholar - 8.El Karoui, N., Rouge, R.: Pricing via utility maximization and entropy. Math. Finance
**10**, 259–276 (2000) MathSciNetCrossRefzbMATHGoogle Scholar - 9.Fleming, W.H., McEneaney, W.M.: Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim.
**33**, 1881–1915 (1995) MathSciNetCrossRefzbMATHGoogle Scholar - 10.Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. de Gruyter, Berlin (2011) CrossRefzbMATHGoogle Scholar
- 11.Frittelli, M., Gianin, E.R.: Putting order in risk measures. J. Bank. Finance
**26**, 1473–1486 (2002) CrossRefGoogle Scholar - 12.Fuhrman, M., Hu, Y., Tessitore, G.: Ergodic BSDEs and optimal ergodic control in Banach spaces. SIAM J. Control Optim.
**48**, 1542–1566 (2009) MathSciNetCrossRefzbMATHGoogle Scholar - 13.Henderson, V.: Valuation of claims on non-traded assets using utility maximization. Math. Finance
**12**, 351–373 (2002) MathSciNetCrossRefzbMATHGoogle Scholar - 14.Henderson, V., Hobson, D.: Utility indifference pricing: an overview. In: Carmona, R. (ed.) Indifference Pricing, pp. 44–73. Princeton University Press, Princeton (2009) Google Scholar
- 15.Henderson, V., Liang, G.: Pseudo linear pricing rule for utility indifference valuation. Finance Stoch.
**18**, 593–615 (2014) MathSciNetCrossRefzbMATHGoogle Scholar - 16.Hu, Y., Imkeller, P., Müller, M.: Utility maximization in incomplete markets. Ann. Appl. Probab.
**15**, 1691–1712 (2005) MathSciNetCrossRefzbMATHGoogle Scholar - 17.Hu, Y., Jin, H., Zhou, X.: Time-inconsistent stochastic linear-quadratic control. SIAM J. Control Optim.
**50**, 1548–1572 (2012) MathSciNetCrossRefzbMATHGoogle Scholar - 18.Hu, Y., Liang, G., Tang, S.: Exponential utility maximization and indifference valuation with unbounded payoffs. Preprint (2018). Available online at arXiv:1707.00199
- 19.Hu, Y., Madec, P., Richou, A.: A probabilistic approach to large time behaviour of mild solutions of HJB equations in infinite dimension. SIAM J. Control Optim.
**53**, 378–398 (2015) MathSciNetCrossRefzbMATHGoogle Scholar - 20.Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1998) CrossRefzbMATHGoogle Scholar
- 21.Klöppel, S., Schweizer, M.: Dynamic indifference valuation via convex risk measures. Math. Finance
**17**, 599–627 (2007) MathSciNetCrossRefzbMATHGoogle Scholar - 22.Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab.
**28**, 558–602 (2000) MathSciNetCrossRefzbMATHGoogle Scholar - 23.Liang, G., Zariphopoulou, T.: Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE. SIAM J. Financ. Math.
**8**, 344–372 (2017) MathSciNetCrossRefzbMATHGoogle Scholar - 24.Mania, M., Schweizer, M.: Dynamic exponential utility indifference valuation. Ann. Appl. Probab.
**15**, 2113–2143 (2005) MathSciNetCrossRefzbMATHGoogle Scholar - 25.Musiela, M., Zariphopoulou, T.: An example of indifference prices under exponential preferences. Finance Stoch.
**8**, 229–239 (2004) MathSciNetCrossRefzbMATHGoogle Scholar - 26.Musiela, M., Zariphopoulou, T.: Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model. In: Fu, M.C., et al. (eds.) Advances in Mathematical Finance, Applied and Numerical Harmonic Analysis, pp. 303–334. Birkhäuser, Boston (2007) Google Scholar
- 27.Musiela, M., Zariphopoulou, T.: Optimal asset allocation under forward exponential performance criteria. In: Ethier, S.N., et al. (eds.) Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, pp. 285–300. Institute of Mathematical Statistics, Ohio (2008) CrossRefGoogle Scholar
- 28.Musiela, M., Zariphopoulou, T.: Derivative pricing, investment management and the term structure of exponential utilities: the case of binomial model. In: Carmona, R. (ed.) Indifference Pricing, pp. 3–41. Princeton University Press, Princeton (2009) Google Scholar
- 29.Musiela, M., Zariphopoulou, T.: Portfolio choice under dynamic investment performance criteria. Quant. Finance
**9**, 161–170 (2009) MathSciNetCrossRefzbMATHGoogle Scholar - 30.Musiela, M., Zariphopoulou, T.: Portfolio choice under space-time monotone performance criteria. SIAM J. Financ. Math.
**1**, 326–365 (2010) MathSciNetCrossRefzbMATHGoogle Scholar - 31.Musiela, M., Zariphopoulou, T.: Stochastic partial differential equations and portfolio choice. In: Chiarella, C., et al. (eds.) Contemporary Quantitative Finance. Essays in Honour of Eckhard Platen, pp. 195–215. Springer, Berlin (2010) CrossRefGoogle Scholar
- 32.Nadtochiy, S., Tehranchi, M.: Optimal investment for all time horizons and Martin boundary of space-time diffusions. Math. Finance
**27**, 438–470 (2017) MathSciNetCrossRefGoogle Scholar - 33.Nadtochiy, S., Zariphopoulou, T.: A class of homothetic forward investment performance processes with non-zero volatility. In: Kabanov, Y., et al. (eds.) Inspired by Finance: The Musiela Festschrift, pp. 475–505. Springer, Berlin (2014) CrossRefGoogle Scholar
- 34.Riedel, F.: Dynamic coherent risk measures. Stoch. Process. Appl.
**112**, 185–200 (2004) MathSciNetCrossRefzbMATHGoogle Scholar - 35.Shkolnikov, M., Sircar, R., Zariphopoulou, T.: Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations. SIAM J. Financ. Math.
**7**, 588–618 (2016) MathSciNetCrossRefzbMATHGoogle Scholar - 36.Zariphopoulou, T., Žitković, G.: Maturity-independent risk measures. SIAM J. Financ. Math.
**1**, 266–288 (2010) MathSciNetCrossRefzbMATHGoogle Scholar - 37.Žitković, G.: A dual characterization of self-generation and exponential forward performances. Ann. Appl. Probab.
**19**, 2176–2210 (2009) MathSciNetCrossRefzbMATHGoogle Scholar

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