Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion

In this paper, we consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in [10.11], we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in [28]. Then we obtain a generalized dynamic programming principle and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.


Introduction
Non-linear BSDEs in the framework of linear expectation were introduced by Pardoux and Peng [18] in 1990. Then a lot of researches were studied by many authors and they provided various applications of BSDEs in stochastic control, finance, stochastic differential games and second order partial differential equations theory, see [1,8,12,13,[19][20][21]27].
The notion of sublinear expectation space was introduced by Peng [14][15][16], which is a generalization of classical probability space. The G-expectation, a type of sublinear expectation, has played an important role in the researches of sublinear expectation space recently. It can be regarded as a counterpart of the Wiener probability space in the linear case. Within this G-expectation framework, the G-Brownian motion is the canonical process. Besides, the notions of the G-martingales and the Itô integral w.r.t. G-Brownian motion were also derived. There are some new structures in these notions and some new applications in the financial models with volatility uncertainty, see Peng [16,17].
In the G-expectation framework, thanks to a series of studies [23][24][25][26], the complete representation theorem for G-martingales has been obtained by Peng, Song and Zhang [22]. Due to this contribution, a natural formulation of BSDEs driven by G-Brownian motion was found by Hu, Ji, Peng and Song [10]. In addition, the existence and uniqueness of the solution to the BSDEs driven by G-Brownian motion has been proved. They also have given the comparison theorem, Feynman-Kac Formula and Girsanov transformation for BSDEs driven by G-Brownian motion in [11]. So the complete theory of BSDEs driven by G-Brownian motion has been established.
An important application of BSDEs is that we can define the recursive utility functions from BSDEs, which can index scaling risks in the study of economics and finance [2,[5][6][7]. Based on these results, a type of significant stochastic optimal control problems under linear expectation with a BSDE as cost function were studied [1,12,20,21,27]. Under G-expectation, the similar problems will be useful in the future studies of finance models with volatility uncertainty. So we arise a natural question: Can we construct the similar results in G-expectation framework. When the complete results about BSDEs driven by G-Brownian motion were established in [10,11], we try to prove the complete results of stochastic optimization theory of BSDEs driven by G-Brownian motion in this paper.
In this paper, we investigate the stochastic optimal control problems with a BSDE driven by G-Brownian motion constructed in [10,11] as cost function. Based on the results in [10,11], we obtain the dynamic programming principle under G-expectation. Besides, the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.
The rest of the paper is organized as follows. In Section 2, we recall the Gexpectation framework and adapt it according to our objective. Besides, we give the related properties of forward and backward stochastic differential equations driven by G-Brownian motion, which will be needed in the sequel sections. In Section 3, the stochastic optimal control problems with a BSDE driven by G-Brownian motion as cost function are investigated and a dynamic programming principle under G-expectation is obtained. In Section 4, The value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

Preliminaries
In this section, we recall the G-expectation framework established by Peng [3,[14][15][16]. Besides, we give some results about forward and backward stochastic differential equations driven by G-Brownian motion, which we need in the following sections. Some details can be found in [10,11].

G-expectation and G-martingales
whereX is an independent copy of X, i.e.,X and X is identically distributed and X is independent from X. Here the letter G denotes the function where S d denotes the collection of all d × d symmetric matrices.
In particular, E[ϕ(X)] = u(1, 0), where u is the unique viscosity solution of the following parabolic PDE defined on [0, ∞) × R d : for any A ≥ B, we call the G-normal distribution non-degenerate, which is the case we consider throughout this paper.
(i) The G-expectationÊ is a sublinear expectation defined bŷ We denote by L p G (Ω T ), p ≥ 1, the completion of G-expectation space L ip (Ω T ) under the norm X p,G := (Ê[|X| p ]) 1 p . For all t ∈ [0, T ],Ê[·] andÊ t [·] are continuous mapping on L ip (Ω T ) endowed with the norm · 1,G . Therefore, it can be extended continuously to L p G (Ω T ).
= M s . Now we introduce the Itô integral and quadratic variation process with respect to G-Brownian motion in G-expectation space.
as the collection of following type of simple processes: The mapping I : is continuous and thus can be continuously extended to M 2 G (0, T ). Definition 2.9. The quadratic variation process of G-Brownian motion is defined by which is a continuous, nondecreasing process.

Definition 2.10.
We now define the integral of a process η ∈ M 1 G (0, T ) with respect to B as following: The mapping is continuous and can be extended to M 1 G (0, T ) uniquely.
Then we detail some results about the quasi-analysis theory constructed in [3].
Theorem 2.11. There exists a weakly compact family P ⊂ M 1 (Ω T ), the collection of probability measures defined on (Ω T , B(Ω T )), such that P is called a set of probability measures that representsÊ.
Definition 2.12. We define the capacity associated to P, which is a weakly compact family of probability measure representsÊ, as follow: c is also called the capacity induced byÊ. (ii) Let X and Y be two random variables, we say that X is a version of Y, if X = Y q.s.
The completion of C b (Ω T ) and L ip (Ω T ) under · p,G are the same and we denote them by L p G (Ω T ).

Forward and Backward Stochastic Differential Equations Driven by G-Brownian Motion
We consider the following stochastic differential equations driven by d dimensional G-Brownian motion (G-SDE): Theorem 2.14. ( [16]) There exists a unique solution X ∈ M 2 G (0, T ; R n ) of the stochastic differential equation (2.5).

Now we give the results about BSDEs driven by G-Brownian motion in the
We consider the following type of G-BSDEs (we always use Einstein convention), For simplicity, we denote by G(0, T ) the collection of processes (Y, Theorem 2.16. ( [10]) Assume that ξ ∈ L β G (Ω T ) and f, g satisfy (H1) and (H2) for some β > 1. Then equation (2.6) has a unique solution (Y, Z, K). Moreover, for . We have the following estimates.

A DPP for Stochastic Optimal Control Problems under G-Expectation
Now we introduce the setting for stochastic optimal control problems under Gexpectation. We suppose that the control state space V is a compact metric space. Let the set of admissible control processes U for the player be a set of V-valued stochastic processes in M β G ([t, T ]; R n ) with β > 2 and t ∈ [0, T ]. For a given admissible control υ(·) ∈ U, the corresponding orbit which regards t as the initial time and ξ ∈ L 2 G (Ω t ; R n ) as the initial state, is defined by the solution of the following type of G-SDE: are deterministic functions and satisfy the following conditions (H3): From the assumption (H3), we can get global linear growth conditions for b, h i j , σ j , i.e., there exists C > 0 such that, for t ∈ [0, T ], x ∈ R n , υ ∈ U, |σ j (t, x, υ)| ≤ C(1 + |x| + |υ|). Obviously, under the above assumptions, for any υ(·) ∈ U, G-SDE (3.1) has a unique solution. Moreover, we have the following estimates : where C depends on L, G, p, n, T .
Proof. The proof is similar to the proof of Proposition 4.1 in [11].

Now we give bounded functions
(ii) For every fixed (x, y, z, υ) ∈ R n ×R×R n ×U, f (·, x, y, z, υ) and g i j (·, x, y, z, υ) are continuous in t, From (H4), we have that Φ, f and g i j also satisfy global linear growth condition in x, i.e., there exists C > 0, such that for all 0 ≤ t ≤ T , υ ∈ U, x ∈ R n , For any υ ∈ U and ξ ∈ L 2 G (Ω t , R n ), the mappings f (s, x, y, z, υ) := f (s, X t,ξ;υ s , y, z, υ s ) and g i j (s, x, y, z, υ) = g i j (s, where X t,ξ;υ is introduced by (3.1).
where C depends on L, G, n and T .
Proof. The proof is similar to the Proposition 4.2 in [11].
Given a control process υ(·) ∈ U, we introduce an associated cost functional where the process Y t,ξ;υ t is defined by G-BSDE (3.2). Similar to the proof of Theorem 4.4 in [11], we have that for t ∈ [0, T ], ξ ∈ L 2 G (Ω t , R n ), But we are more interest in the case when ξ = x. Now we define the value function as follow:

Proposition 3.3. u(t, x) is a deterministic function of (t, x).
Proof. For a partition of [t, s]: When υ(s) ∈ M p,t G (t, s; R n ), we note that J(t, x; υ) is a deterministic function of (t, x), because b, h i j , σ j , Φ, f and g i j are deterministic functions and B s := B t+s − B t is a G-Brownian motion. So we need to construct a sequence of admissible controls {υ i (·)} of the form x; υ k ). Then we define υ, υ ′ ∈ U, Therefore, .

Now we suppose that
HenceÊ[u(t, x)] = u(t, x). We have finished the proof.
Conversely, ∀ε > 0, there exists a υ(·) ∈ U, such that Proof. We already know that u(t, x) is continuous with respect to x and Y t,ζ;υ t is continuous with respect to (ζ, υ(·)). We want to prove (3.6), only need to discuss the simple random variables ζ of the form and υ(·) of the form Here i = 1, 2, ..., N, is a B(Ω t )-partition. Then from the same technique used in the proof of Theorem 4.4 in [11], we have So we have proved (3.6). Now we prove (3.7) in a similar way. We first construct a random variable η ∈ L 2 G (Ω t ; R n ), for υ(·) ∈ U. Now, we chose a control υ i (·) ∈ M p,t G (t, s; R n ), such that u(t, So we have (3.7). Now we give a type of DPP for our stochastic optimal control problems. Firstly, we define a family of backward semigroups associated with the G-BSDE (3.2). Given the initial data (t, x), a positive number δ ≤ T − t and a random variable η ∈ L p G (Ω; R) with p > 1, we set where (Y t,x;υ s ) t≤s≤t+δ is the solution of the following G-BSDE with the time horizon t + δ: Obviously, for the solution Y t,x;υ · of G-BSDE (3.2), we have Then we can obtain the DPP for our stochastic optimal control problems as follow: Besides, for ε > 0, there exists an admissible controlῡ(·) ∈ U such that Then Because ε can be arbitrarily small, we get (3.8).
Proposition 3.7. u(t, x) is 1 2 -Hölder continuous in t. Proof. For any given (t, x) ∈ [0, T ] × R n and δ > 0(t + δ ≤ T ), from Theorem 3.6, we know that for ε > 0, there exists a υ(·) ∈ U such that Then we need to prove We only check the first inequality in (3.9). The second can be proved similarly.