Backward Stochastic Equations Research Articles

On optimal control of coupled mean-field forward-backward stochastic equations

Abstract We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type.

Observability inequality of backward stochastic heat equations with Lévy process for measurable sets and its applications

This paper endeavours to directly establish the observability inequality of backward stochastic heat equations involving a Lévy process for measurable sets. As an immediate application, we attain the approximate controllability of forward stochastic heat equations featuring a Lévy process. Subsequently, we embark upon the formulation of a nuanced optimization problem, encompassing both the determination of an optimal actuator location and its associated minimum norm control. This formulation is elegantly cast as a two-person zero-sum game problem. We then proceed to articulate a set of conditions–both sufficient and necessary–required for optimizing the solution through the prism of a Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.

Open-loop and closed-loop Nash equilibria for the LQ stochastic difference game

This paper investigates the conditions for the existence of open-loop and closed-loop Nash equilibria for the linear quadratic (LQ) two-player stochastic difference game. It is shown that the former is characterized by the solvability of a forward–backward stochastic difference equation (FBSΔE), while the latter is described by the solvability of a class of coupled symmetric difference Riccati equations (ΔREs). Different from the existing works, the relationship between the closed-loop representation of open-loop Nash equilibrium and the closed-loop Nash equilibrium has been studied. It is found that the closed-loop representation of open-loop Nash equilibrium is different from the outcome of closed-loop Nash equilibrium for the general case. However, for the LQ zero-sum stochastic difference game, they are consistent.

First- and Second-Order Maximum Principles for Discrete-Time Stochastic Optimal Control With Recursive Utilities

This paper deals with the discrete-time stochastic optimal control problems with recursive utilities under weakened convexity assumption. A new stochastic maximum principle is established. Moreover, by constructing two new adjoint equations and two new variational equations for the backward stochastic difference equation (BS <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta$</tex-math></inline-formula> E), we obtain the second-order necessary optimality condition of quasi-singular control. Finally, as an illustration, a discrete-time mean-variance portfolio selection mixed with a recursive utility functional optimization problem is solved.

Exact controllability of forward and backward stochastic difference system

In this paper, we study the exact controllability of forward and backward stochastic difference system (FBSDS) with multiplicative noise. Different from the standard system governed by forward stochastic difference equation (FSDE), the inclusion of both initial and terminal conditions in FBSDS complicates the solvability which also leads to the challenge of exact controllability. To overcome the difficulty, we transform the FBSDS into an equivalent backward stochastic difference equation. Accordingly, we derive both the necessary and sufficient Gramian matrix criterion and the Rank criterion for the exact controllability of FBSDS, which is the main contribution of the paper.

Optimal Controls of Stochastic Differential Equations with Jumps and Random Coefficients: Stochastic Hamilton–Jacobi–Bellman Equations with Jumps

We study the stochastic Hamilton–Jacobi–Bellman (HJB) equation with jump, which arises from a non-Markovian optimal control problem with a recursive utility cost functional. The solution to the equation is a predictable triplet of random fields. We show that the value function of the control problem, under some regularity assumptions, is the solution to the stochastic HJB equation; and a classical solution to this equation is the value function and characterizes the optimal control. With some additional assumptions on the coefficients, an existence and uniqueness result in the sense of Sobolev space is shown by recasting the stochastic HJB equation as a backward stochastic evolution equation in Hilbert spaces with the Brownian motion and Poisson jump.

First-order and second-order necessary optimality conditions for discrete-time stochastic systems with delay

Abstract This paper is concerned with the first-order and second-order necessary optimality conditions for discrete-time stochastic systems with delay under weak assumptions. Based on a characteristic method and a discrete-time backward stochastic equation, we establish the linearized discrete-time stochastic maximum principle and Euler-type necessary optimality condition. Moreover, by a new discrete-time backward stochastic matrix equation, we also obtain the necessary optimality condition of quasi-singular control and the pointwise second-order optimality condition.

Maximum principle for stochastic optimal control problem of finite state forward‐backward stochastic difference systems

AbstractThis article studies the maximum principle for stochastic optimal control problems of forward‐backward stochastic difference systems (FBSSs) where the uncertainty is modeled by a discrete time, finite state process, rather than white noises. Two distinct forms of FBSSs are investigated. The first one is described by a partially coupled forward‐backward stochastic difference equation (FBSE) and the second one is described by a fully coupled FBSE. We deduce the adjoint difference equation by adopting an appropriate representation of the product rule and a proper formulation of the backward stochastic difference equation (BSE). Finally, the maximum principle for this optimal control problem with the convex control domain is established.

Martingale Decomposition and Backward Stochastic Dynamic Equations on Time Scales

The paper aims to establish the related backward stochastic dynamic equations on time scales, BS ∇ Es for short, concerning to ∇‐integral on time scales. We present the martingale decomposition theorem on time scales and prove the existence and uniqueness theorem of solutions to BS ∇ Es. This work can be considered as a unification and a generalization of similar results in backward stochastic difference equations and backward stochastic differential equations.

Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity

In this paper, we consider a null controllability and an inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. We construct two special weight functions to establish two Carleman estimates for the whole stochastic Grushin operator with singular potential by a weighted identity method. One is for the backward stochastic Grushin equation with singular weight function. We then apply it to prove the null controllability for stochastic Grushin equation for any T and any degeneracy γ &gt; 0, when our control domain touches the degeneracy line {x = 0}. In order to study the inverse source problem of determining two kinds of sources simultaneously, we prove the other Carleman estimate, which is for the forward stochastic Grushin equation with regular weight function. Based on this Carleman estimate, we obtain the uniqueness of the inverse source problem.

SDEs with random and irregular coefficients

We consider Itô uniformly nondegenerate equations with random coefficients. When the coefficients satisfy some low regularity assumptions with respect to the spatial variables and Malliavin differentiability assumptions on the sample points, the unique solvability of singular SDEs is proved by solving backward stochastic Kolmogorov equations and utilizing a modified Zvonkin type transformation.

Forward-backward stochastic equations: a functional fixed point approach

We introduce forward-backward stochastic equations (FBSEs) that incorporate fully-coupled forward-backward structure into backward stochastic equations (BSEs) introduced in Cheridito and Nam (Ann. Probab. 45(6A):3795–3828, 2017). Such a system generalizes the classical backward stochastic differential equations (BSDEs) and forward-backward stochastic differential equations (FBSDEs) in previous literature. We transform an FBSE into a fixed point equation on the space of random variables and then apply general fixed point theorems to derive the existence and/or uniqueness of a solution. As a result, we obtain novel existence and/or uniqueness results for fully-coupled FBSDEs with functional drivers, which are either Lipschitz or non-Lipschitz.

Strong error estimates for a space-time discretization of the linear-quadratic control problem with the stochastic heat equation with linear noise

Abstract We propose a time-implicit, finite-element-based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+\sigma (t)]\,\,\textrm{d}W(t)$ and prove optimal convergence w.r.t. both space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work by Prohl &amp; Wang (2021, Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation. ESAIM Control Optim. Calc. Var., 27, 54), where the error analysis was restricted to additive noise.

First-order and second-order necessary optimality conditions concerning components for discrete-time stochastic systems

This paper deals with the first-order and second-order necessary optimality conditions concerning the components for discrete-time stochastic systems under weakened convexity assumptions. By means of a new discrete-time backward stochastic equation, we establish a first-order necessary optimality condition concerning every component in the form of a global stochastic maximum principle. Besides, we also obtain an optimality condition concerning one component of vector control in the form of a global stochastic maximum principle and concerning the other in the form of linearised stochastic maximum principle. Moreover, by introducing a new discrete-time backward stochastic matrix equation, we derive diverse second-order necessary optimality conditions of singular and quasi-singular controls concerning the components.

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.

Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation

We verify strong rates of convergence for a time-implicit, finite-element based space-time discretization of the backward stochastic heat equation, and the forward-backward stochastic heat equation from stochastic optimal control. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise. This work is thus giving a theoretical foundation for the computational findings in Dunst and Prohl, SIAM J. Sci. Comput. 38 (2016) A2725–A2755.

A maximum principle for fully coupled controlled forward\u2013backward stochastic difference systems of mean-field type

In this paper, we consider the optimal control problem for fully coupled forward–backward stochastic difference equations of mean-field type under weak convexity assumption. By virtue of employing a suitable product rule and formulating a mean-field backward stochastic difference equation, we establish the stochastic maximum principle and also derive, under additional assumptions, that the stochastic maximum principle is also a sufficient condition. As an application, a Stackelberg game of mean-field backward stochastic difference equation is presented to demonstrate our results.

L2-regularity of solutions to linear backward stochastic heat equations, and a numerical application

In this work, we mainly explore the L2-regularity for the second component of the solutions to linear backward stochastic heat equations, which is crucial to obtain the convergence of the numerical solutions. As an application, we provide the convergence rate for time-discretized Galerkin approximation of these equations.

Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

Let (Ω,F,F,P) be a filtered probability space with a filtration F=(Ft)t∈[0,T] satisfying the usual conditions and T a finite time, L0(F0) the algebra of equivalence classes of F0–measurable real–valued random variables on Ω, Lp(FT,Rd) the usual function space and LF0p(FT,Rd) the L0(F0)–module generated by Lp(FT,Rd). The usual backward stochastic equations (BSEs) are studied for their terminal conditions ξ in Lp(FT,Rd). Motivated by the study of continuous–time conditional mean–conditional convex risk measure portfolio selection, this paper, for the first time, formulates and studies a more general class of BSEs with their terminal conditions in LF0p(FT,Rd). The main results of this paper are to prove two fixed point theorems in complete random normed modules, which are respectively the random generalizations of Banach contraction mapping principle and Browder–Kirk fixed point theorem, and give their applications to the general class of BSEs.

The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth

The paper is devoted to the Cauchy problem of backward stochastic super-parabolic equations with quadratic growth. We prove two Itô formulas in the whole space. Furthermore, we prove the existence of weak solutions for the case of one-dimensional state space, and the uniqueness of weak solutions without constraint on the state space.