An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

1 Introduction

In this paper, we aim, using Pontryagin’s maximum principle, to prove necessary as well as sufficient optimality conditions, for a risk-sensitive control problem associated with dynamics driven by a backward doubly stochastic differential equation (BDSDE in short). We solve the problem by using the approach developed by Djehiche, Tembine and Tempone in [5]; see also [2]. Our contribution can be summarized as follows: Djehiche et al. [5] have established a stochastic maximum principle for a class of risk-sensitive mean-field type control problems, where the distribution enters only through the mean of a state process, which means that the drift, diffusion, and terminal cost functions depend on the state, the control and the means of the state process. Their work extends the results of Lim and Zhou [10] to risk-sensitive control problems for dynamics that are non-Markovian and without mean-field term.

Nonlinear BDSDEs have been introduced by Pardoux and Peng [12]. They have considered a new kind of BSDE that is a class of BDSDEs with two different directions of stochastic integrals, i.e., the equations involve both a standard (forward) stochastic Itô integral d⁢Wt and a backward stochastic Itô integral d⁢B←t. More precisely, they dealt with the following BDSDE:

Here we regard (1.1) as a backward equation in two different aspects. The first is with respect to B for which the time variable is reversed. The second is with respect to W, which is forward in time while the boundary condition is given at the terminal instead of the initial time. They proved that if f and g are uniform Lipschitz, then (1.1) has a unique solution (Yt,Zt) in the interval [0,T], for any Zt being square integrable, and the terminal condition ξ is a ℱTW-measurable and square integrable random variable. They also showed that BDSDEs can produce a probabilistic representation for solutions to some quasi-linear stochastic partial differential equations. Since this is the first existence and uniqueness result, many papers have been devoted to existence and/or uniqueness results under weaker assumptions; see [12] and the references therein. Further, the control processes are assumed to be adapted to a subfiltration of the filtration generated by both forward and backward filtration. In the risk-sensitive control problem, the system is governed by the nonlinear BDSDE

where f, g are given maps, B=(Bt)t≥0 and W=(Wt)t≥0 are two mutually independent standard Brownian motion processes, zt is square integrable and the terminal condition ξ is a ℱTW-measurable and square integrable random variable. The control variable v=(vt) is called admissible control. We denote by 𝒰 the class of all admissible controls.

We define the criterion to be minimized, with initial risk-sensitive functional cost, as follows:

where Ψ and l are given maps and (yt,zt) is the trajectory of the system controlled by vt.

A control u is called optimal if it solves Jθ⁢(u)=infv∈𝒰⁡Jθ⁢(v).

The existence of an optimal solution for this problem has been solved to achieve the objective of this paper, and we have established necessary as well as sufficient optimality conditions for this model. In this paper, we generalize the results obtained by Chala in [2] to the BDSDE. The idea here is to reformulate in the first step the risk-sensitive control problem in terms of an augmented state process and terminal payoff problem. An intermediate stochastic maximum principle (SMP in short) is then obtained by applying the SMP of [1, 8] for a loss functional without running cost; for the same particular cases see [7]. Then we transform the intermediate adjoint processes to a simpler form, using the fact that the set of controls is convex. Then we establish necessary as well as sufficient optimality conditions by using the Logarithmic transform established by El Karoui and Hamadène [6]. More precisely, the method of Lim and Zhou [10] shows in fact that it is enough to use a generic square integrable martingale to transform the pair (p1,q1) into the adjoint process (p~1⁢(t),0), where the process p~1⁢(t) is still a square integrable martingale, which would mean that p~1⁢(t)=p~1⁢(T), and is equal to the constant 𝔼⁢[p~1⁢(T)]. But this generic martingale need not be related to the adjoint process p→⁢(t) as in Lim and Zhou [10]. Instead, it will be part of the adjoint equation associated with the risk-sensitive SMP (see Proposition 4.2 below).

We note that necessary as well as sufficient optimality conditions for risk-sensitive controls, where the systems are governed by a stochastic differential equation (SDE in short), have been studied by Lim and Zhou [10]. We also note that necessary as well as sufficient optimality conditions for stochastic controls, where the systems are governed by a nonlinear forward stochastic differential equation with jumps, have been studied by Shi and Wu in [14] in the case where the set of admissible controls is convex, and in [15] in the general case with an application to finance. Furthermore, the systems governed by a mean-field SDE have been studied by Djehiche et al. in [5]. For some general case, we can note the paper of Khallout and Chala [9] with the fully coupled forward backward stochastic differential equation, and where the admissible control set could be convex.

Our goal in this paper is to establish a set of necessary as well as sufficient optimality conditions for BDSDE with risk-sensitive performance cost. The backward doubly problem under consideration is not simple extension from the mathematical point of view, but also provides interesting models in many applications such as mathematical economics or optimal control. The proof of our main result is based on the spike variation method based on Theorem 3.2 as a preliminary step for the risk-neutral control problem.

The paper is organized as follows: In Section 2, we give the precise formulation of the problem; we introduce the risk-sensitive model, formulate the problem and give the various assumptions used throughout this paper. In Section 3, we shall study our system of BDSDE, and give the relation between the risk-neutral and risk-sensitive SMP. In Section 4, we give the new method of the transformation of the adjoint processes; the aim of this result will be shown in that section. We give our first main result, the necessary optimality conditions for risk-sensitive control problem under additional hypothesis, and the second result will be given in the following section. In Section 5, we derive our second main result of this paper, which are sufficient optimality conditions for risk-sensitive controls. In Section 6, we improve the quality of the paper by giving two applications to a linear quadratic stochastic control problem, the method which used the Riccati equations is applied in a second example, and we give a numerical example. Section 7 consists of a conclusion and an outlook.

2 Formulation of the problem

Let (Ω,ℱ,ℙ) be a probability space in which one-dimensional Brownian motions W=(Wt:0≤t≤T) and B=(Bt:0≤t≤T) are defined, where W and B are two mutually independent standard Brownian motion processes. Let 𝒩 denote the class of ℙ-null sets of ℱ. For each t∈[0,T], we define ℱt(W,B)=ℱtW∨ℱt,TB, where for any process {Lt}t∈[0,T] one has ℱs,tL=σ⁢{Lr-Ls:s≤r≤t}∨𝒩 and ℱtL=ℱ0,tL.

Note that the collection {ℱt(W,B):t∈[0,T]} is neither increasing nor decreasing, and it does not constitute a filtration. We may define the subfiltration (𝒢t)t∈[0,T] such that 𝒢t⊂ℱt(W,B) for all t∈[0,T].

Let ℳ2⁢([0,T];ℝ) denote the set of one-dimensional jointly measurable random processes {φt:t∈[0,T]} which satisfy the following conditions:

1. ∥φ∥ℳ2=𝔼⁢[∫0T|φt|2⁢𝑑t]<∞.

2. φt is ℱt(W,B)-measurable for any t∈[0,T].

Similarly, we denote by 𝒮2⁢([0,T];ℝ) the set of one-dimensional continuous random processes which satisfy the following conditions:

1. ∥φ∥𝒮2=𝔼⁢[supt∈[0,T]⁡|φt|2]<∞.

2. φt is ℱt(W,B)-measurable for any t∈[0,T].

Let T be a positive real number and let 𝒜 be a nonempty subset of ℝ.

For any v∈𝒰, we consider the following BDSDE system:

where f:[0,T]×ℝ×ℝ×𝒜→ℝ, g:[0,T]×ℝ×ℝ×𝒜→ℝ are jointly measurable and such that for any (y,z,v)∈ℝ×ℝ×𝒜 one has f⁢(⋅,y,z,v)∈ℳ2⁢([0,T]×ℝ), g⁢(⋅,y,z,v)∈ℳ2⁢([0,T]×ℝ), and zt is square integrable and the terminal condition ξ is a ℱTW-measurable and square integrable random variable.

Note that the integral with respect to (Bt)t∈[0,T] is a “backward” Itô integral, while the integral with respect to (Wt)t∈[0,T] is a standard forward Itô integral. These two types of integrals are particular cases of the Itô–Skorohod integral; for more details we refer to [11].

We define the criterion to be minimized, with initial risk-sensitive performance functional cost, as

where Ψ:ℝ→ℝ and l:[0,T]×ℝ×ℝ×𝒜→ℝ are jointly measurable and θ is the risk-sensitive index.

The optimal control problem is to minimize the functional Jθ over 𝒰 if u∈𝒰 is an optimal control (solution), that is,

We assume the following.

A control that solves problem (2.1)–(2.3) is called optimal. Our objective is to establish risk-sensitive necessary and sufficient optimality conditions, satisfied by a given optimal control, in the form of risk-sensitive SMP.

We also assume the following.

Under the above Assumptions 2.2 and 2.4, for each v∈𝒰, equation (2.1) has a unique strong solution, and the cost function Jθ is well defined from 𝒰 into ℝ.

For more details the reader can see paper of Han, Peng and Wu in [8].

3 Risk-sensitive stochastic maximum principle of backward doubly type control

The proof of our risk-sensitive stochastic maximum principle necessitates a certain auxiliary state process xt, which is the solution of the following forward SDE:

Our control problem of (2.1)–(2.3) is equivalent to

We require the following condition:

If we put ΘT=Ψ⁢(y0)+∫0Tl⁢(t,yt,zt,vt)⁢𝑑t, then the risk-sensitive loss functional is given by

When the risk-sensitive index θ is small, the loss functional ℋ⁢(θ,u) can be expanded as

where Var⁡(ΘT) denotes the variance of ΘT. If θ<0, the variance of ΘT, as a measure of risk, improves the performance ℋ⁢(θ,v), in which case the optimizer is called risk seeker. But when θ>0, the variance of ΘT worsens the performance ℋ⁢(θ,v), in which case the optimizer is called risk averse. The risk-neutral loss functional 𝔼⁢(ΘT) can be seen as a limit of the risk-sensitive functional ℋ⁢(θ,v) when θ→0; for more details see [4].

Next, let us introduce the following notations.

We suppose that Assumptions 2.2 and 2.4 hold. We may combine the SMP for a risk-neutral controlled differential equation of backward doubly stochastic type from [1, 8] with the result of Yong [16] and with augmented state dynamics (x,y,z) to derive the adjoint equation. There exist two unique 𝒢t-adapted pairs of processes (p1,q1) and (p2,q2) which solve the following system of backward SDEs:

with

where

and

such that

Let H~θ be the Hamiltonian associated with the optimal state dynamics (xu,yu,zu), and let the two pairs of adjoint process ((p1,q1),(p2,q2)) be given by

4 Finding the new adjoint equations and necessary optimality conditions

As mentioned, Theorem (3.2) is a good SMP for the risk-neutral control problem of forward-backward doubly type. We follow the same approach used in [2, 9], and suggest a transformation of the adjoint processes (p1,q1) and (p2,q2) in such a way that we can omit the first component (p1,q1) in (3.3) and express the SMP in terms of only one adjoint process which we denote by (p~2,q~2).

By noting that d⁢p1⁢(t)=q1⁢(t)⁢d⁢Wt and p1⁢(T)=θ⁢ATθ, the explicit solution of this backward SDE is

where Vtθ:=𝔼[ATθ∣𝒢t], 0≤t≤T.

In view of (4.1), it would be natural to choose a transformation of (p→,q→) into an adjoint process (p~,q~), where

We consider the following transform:

By using (3.3) and (4.2), we have

The following properties of the generic martingale Vθ are essential in order to investigate the properties of these new process (p~⁢(t),q~⁢(t)): The process Λθ is the first component of the 𝒢t-adapted pair of processes (Λθ,N) which is the unique solution to the following quadratic backward SDE:

where 𝔼⁢[∫0T|N⁢(t)|2⁢𝑑t]<∞.

Next, we will state and prove the necessary optimality conditions for the system driven by a BDSDE with a risk-sensitive performance functional type. To this end, let us summarize and prove some lemmas that we will use thereafter.