Backward Stochastic Evolution Equations Research Articles

Linear quadratic optimal control problems of infinite‐dimensional mean‐field type with jumps

AbstractIn this paper, we formulate and investigate a framework for the theory of the linear quadratic optimal control problem (LQ problem) for infinite‐dimensional mean‐field stochastic evolution systems with jumps. We ensure the well‐posedness of the investigated problems by establishing the existence, uniqueness, and a priori estimates for mild solutions to general infinite‐dimensional mean‐field forward stochastic evolution equations (MF‐SEE) and mean‐field backward stochastic evolution equations (MF‐BSEE) with jumps. Leveraging the Yosida approximation theory, we establish a dual theory between MF‐SEE and MF‐BSEE with jumps, overcoming the challenge posed by the inapplicability of Itô's formula in the context of mild solutions. Our main results regarding the existence and uniqueness of the optimal control, along with corresponding dual characterizations and state feedback representations, are obtained through convex analysis techniques, our established dual theory, and decoupling methods.

Forward–backward stochastic evolution equations in infinite dimensions and application to LQ optimal control problems

This paper focuses on the study of forward–backward stochastic evolution equations (FBSEEs), which are a class of nonlinear fully coupled forward–backward stochastic differential equations (FBSDEs), in infinite dimensions. Drawing inspiration from various linear-quadratic (LQ) optimal control problems, we apply a set of domination-monotonicity conditions that are more relaxed compared to general conditions. Within this framework, we employ the method of continuation to establish the unique solvability result and provide a pair of solution estimates. Notably, to address the challenges posed by the infinite-dimensional setting, we introduce a class of approximating equations, as Itô’s formula is not directly applicable. Conversely, these results find application in various LQ problems, where the stochastic Hamiltonian systems precisely correspond to the FBSEEs satisfying the aforementioned domination-monotonicity conditions. Consequently, by solving the corresponding stochastic Hamiltonian systems, we can obtain explicit expressions for the optimal controls.

Duality between controllability and observability of stochastic linear evolution equations with general filtration

This article is devoted to establishing a general duality for controllability and observability of forward and backward stochastic linear evolution equations with general filtration. The main tool that we employ is the stochastic transposition method introduced in [13]. First, we prove the well-posedness of controlled forward and backward stochastic linear evolution equations with unbounded control operators. Then, we obtain the equivalence between controllability of stochastic evolution equations and observability for their dual equations. Finally, an example is given for illustrating our general results.

Stochastic optimal control with random coefficients and associated stochastic Hamilton\u2013Jacobi\u2013Bellman equations

We consider the optimal control problem for stochastic differential equations (SDEs) with random coefficients under the recursive-type objective functional captured by the backward SDE (BSDE). Due to the random coefficients, the associated Hamilton–Jacobi–Bellman (HJB) equation is a class of second-order stochastic PDEs (SPDEs) driven by Brownian motion, which we call the stochastic HJB (SHJB) equation. In addition, as we adopt the recursive-type objective functional, the drift term of the SHJB equation depends on the second component of its solution. These two generalizations cause several technical intricacies, which do not appear in the existing literature. We prove the dynamic programming principle (DPP) for the value function, for which unlike the existing literature we have to use the backward semigroup associated with the recursive-type objective functional. By the DPP, we are able to show the continuity of the value function. Using the Itô–Kunita’s formula, we prove the verification theorem, which constitutes a sufficient condition for optimality and characterizes the value function, provided that the smooth (classical) solution of the SHJB equation exists. In general, the smooth solution of the SHJB equation may not exist. Hence, we study the existence and uniqueness of the solution to the SHJB equation under two different weak solution concepts. First, we show, under appropriate assumptions, the existence and uniqueness of the weak solution via the Sobolev space technique, which requires converting the SHJB equation to a class of backward stochastic evolution equations. The second result is obtained under the notion of viscosity solutions, which is an extension of the classical one to the case for SPDEs. Using the DPP and the estimates of BSDEs, we prove that the value function is the viscosity solution to the SHJB equation. For applications, we consider the linear-quadratic problem, the utility maximization problem, and the European option pricing problem. Specifically, different from the existing literature, each problem is formulated by the generalized recursive-type objective functional and is subject to random coefficients. By applying the theoretical results of this paper, we obtain the explicit optimal solution for each problem in terms of the solution of the corresponding SHJB equation.

Infinite Horizon Stochastic Maximum Principle for Stochastic Delay Evolution Equations in Hilbert Spaces

In the present work, we investigate infinite horizon optimal control problems driven by a class of stochastic delay evolution equations in Hilbert spaces and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation (ABSEE). We first establish a priori estimate for the solution to ABSEEs by imposing restriction on unbounded operator $$A^*$$ , that is, the operator $$A^*$$ is maximal dissipative. In this way, the Ito inequality is applicable in our study, and we can also avoid the problem that neither Ito formula nor energy equation is available. Next, we obtain the existence and uniqueness results of solutions of linear backward stochastic evolution equations on infinite horizon by using approximating methods. Then, the existence and uniqueness results of ABSEEs on infinite horizon is obtained via the fixed-point theory. That is the highlight of innovation in this paper. Eventually, we establish necessary and sufficient conditions for optimality of the control problem on infinite horizon, in the form of Pontryagin’s maximum principle.

Second Order Necessary Conditions for Optimal Control Problems of Stochastic Evolution Equations

In this paper, we obtain some second order necessary conditions for optimal control problems of stochastic evolution equations in infinite dimensions. The control acts on both the drift and diffusion terms while the control region is assumed to be convex. The concepts of relaxed and $V$-transposition solutions (introduced in our previous works) to operator-valued backward stochastic evolution equations are employed to formulate these optimality conditions. The correction part of the second order adjoint process, which does not appear in the Pontryagin-type stochastic maximum principle, plays a fundamental role in our results.

First and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraints

The purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints. The control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable Hilbert space. We employ some classical set-valued analysis tools and theories of the transposition solution of vector-valued backward stochastic evolution equations and the relaxed-transposition solution of operator-valued backward stochastic evolution equations to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the first order optimality condition, plays a fundamental role in the second order optimality condition.

Optimal Control for Stochastic Delay Evolution Equations

In this paper, we investigate a class of infinite-dimensional optimal control problems, where the state equation is given by a stochastic delay evolution equation with random coefficients, and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation. We first prove the continuous dependence theorems for stochastic delay evolution equations and anticipated backward stochastic evolution equations, and show the existence and uniqueness of solutions to anticipated backward stochastic evolution equations. Then we establish necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin's maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study two examples, an infinite-dimensional linear-quadratic control problem with delay and an optimal control of a Dirichlet problem for a stochastic partial differential equation with delay. Further applications of the two examples to a Cauchy problem for a controlled linear stochastic partial differential equation and an optimal harvesting problem are also considered.

Transposition method for backward stochastic evolution equations revisited, and its application

The main purpose of this paper is to improve our transposition method to solve both vector-valued and operator-valued backward stochastic evolution equations with a general filtration. As its application, we obtain a general Pontryagin-type maximum principle for optimal controls of stochastic evolution equations in infinite dimensions. In 1articular, we drop the technical assumption appeared in [Q. L\"u and X. Zhang, Springer Briefs in Mathematics,Springer, New York, 2014, Theorem 9.1].

Mean-Field Backward Stochastic Evolution Equations in Hilbert Spaces and Optimal Control for BSPDEs

We obtain the existence and uniqueness result of the mild solutions to mean-field backward stochastic evolution equations (BSEEs) in Hilbert spaces under a weaker condition than the Lipschitz one. As an intermediate step, the existence and uniqueness result for the mild solutions of mean-field BSEEs under Lipschitz condition is also established. And then a maximum principle for optimal control problems governed by backward stochastic partial differential equations (BSPDEs) of mean-field type is presented. In this control system, the control domain need not to be convex and the coefficients, both in the state equation and in the cost functional, depend on the law of the BSPDE as well as the state and the control. Finally, a linear-quadratic optimal control problem is given to explain our theoretical results.

Wiener Chaos Solutions for Linear Backward Stochastic Evolution Equations

Wiener chaos expansions have been used by Lototsky and Rozovskii for constructing solutions to forward stochastic evolution equations. In this work we propose a similar methodology for studying linear backward stochastic evolution equations (BSEEs) and introduce generalized solutions of BSEEs in weighted Wiener chaos spaces.

Sufficient conditions of optimality for backward stochastic evolution equations

Sufficient conditions of optimality for backward stochastic evolution equations

ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS

In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that contains backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasi-linear stochastic equations, which includes the reaction diffusion and porous medium equations.

Successive approximation of infinite dimensional semilinear backward stochastic evolution equations with jumps

In this paper, we study the existence and uniqueness of mild solutions to semilinear backward stochastic evolution equations driven by the cylindrical I -Brownian motion and the Poisson point process in a Hilbert space with non-Lipschitzian coefficients by the successive approximation.

On backward stochastic evolution equations in Hilbert spaces and optimal control

In this paper a new result on the existence and uniqueness of the adapted solution to a backward stochastic evolution equation in Hilbert spaces under a non-Lipschitz condition is established. The applicability of this result is then illustrated in a discussion of a concrete backward stochastic partial differential equation. Furthermore, a stochastic maximum principle for optimal control problems of stochastic systems governed by backward stochastic evolution equations in Hilbert spaces is obtained.

Approximate controllability of backward stochastic evolution equations in Hilbert spaces

In this paper, we examine the approximate controllability of a semilinear backward stochastic evolution equations in Hilbert spaces with non-Lipschitz coefficient.

Strong, mild and weak solutions of backward stochastic evolution equations

Four types of solutions ( mild, weak, weak mild and strong ) of the so-called backward stochastic evolution equations in infinite dimensions are proved to exist. A relationship between these solutions is established as well.

Strong, mild and weak solutions of backward stochastic evolution equations

Strong, mild and weak solutions of backward stochastic evolution equations