Coupled Forward-backward Stochastic Differential Equations Research Articles

Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Differential Equations Driven by Subdiffusion

Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Differential Equations Driven by Subdiffusion

A Strong Averaging Principle Rate for Two-Time-Scale Coupled Forward–Backward Stochastic Differential Equations Driven by Fractional Brownian Motion

A Strong Averaging Principle Rate for Two-Time-Scale Coupled Forward–Backward Stochastic Differential Equations Driven by Fractional Brownian Motion

Numerical Method for Multi-Dimensional Coupled Forward-Backward Stochastic Differential Equations Based on Fractional Fourier Fast Transform

Forward-backward stochastic differential equations (FBSDEs) have received more and more attention in the past two decades. FBSDEs can be applied to many fields, such as economics and finance, engineering control, population dynamics analysis, and so on. In most cases, FBSDEs are nonlinear and high-dimensional and cannot be obtained as analytic solutions. Therefore, it is necessary and important to design their numerical approximation methods. In this paper, a novel numerical method of multi-dimensional coupled FBSDEs is proposed based on a fractional Fourier fast transform (FrFFT) algorithm, which is used to compute the Fourier and inverse Fourier transforms. For the forward component of FBSDEs, time discretization is used as well as the backward equation to yield a recursive system with terminal conditions. For the numerical experiments to be successful, three types of numerical methods were used to solve the problem, which ensured the efficiency and speed of computation. Finally, the numerical methods for different examples are verified.

Coupled FBSDEs with measurable coefficients and its application to parabolic PDEs

Using purely probabilistic methods, we prove the existence and the uniqueness of solutions for a system of coupled forward-backward stochastic differential equations (FBSDEs) with measurable, possibly discontinuous, coefficients. As a corollary, we obtain the well-posedness of semilinear parabolic partial differential equations (PDEs)Lu(t,x)+F(t,x,u,∂xu)=0;u(T,x)=h(x) in the natural domain of the second order linear parabolic operator L when F and h are not necessarily continuous with respect to x. We also provide a sufficient condition for this solution to be in a Sobolev space. Finally, we apply the result to optimal policymaking for pandemics and pricing of carbon emission allowances.

Existence of an optimal control for a system driven by a degenerate coupled forward–backward stochastic differential equations

We establish the existence of an optimal control for a system driven by a coupled forward–backward stochastic differential equation (FBDSE) whose diffusion coefficient may degenerate (i.e. are not necessary uniformly elliptic). The cost functional is defined as the initial value of the backward component of the solution. We construct a sequence of approximating controlled systems, for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we get the existence of a feedback optimal control. Filippov's convexity condition is used to ensure that the optimal control is strict. The present result extends those obtained in [2,4] to controlled systems of coupled SDE–BSDE.

Stochastic Differential Games for Fully Coupled FBSDEs with Jumps

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. For SDGs, the upper and the lower value functions are defined by the controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in [6], we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. Furthermore, for a special case (when $\sigma,\ h$ do not depend on $y,\ z,\ k$), under the Isaacs' condition, we get the existence of the value of the game.

Arrow Sufficient Conditions for Optimality of Fully Coupled Forward–Backward Stochastic Differential Equations with Applications to Finance

This paper is concerned with optimal control problems of fully coupled forward---backward stochastic differential equations on finite horizon and infinite horizon with partial information. Two sufficient conditions for optimality are established for the above problems. We demonstrate their applications by four illustrative examples in the framework of cash management, risk minimizing, and linear-quadratic optimal control problems. These examples are explicitly solved based on the sufficient conditions and the optimal filtering of forward---backward stochastic differential equations derived in this paper.

Stochastic Maximum Principle of Near-Optimal Control of Fully Coupled Forward-Backward Stochastic Differential Equation

This paper first makes an attempt to investigate the near-optimal control of systems governed by fully nonlinear coupled forward-backward stochastic differential equations (FBSDEs) under the assumption of a convex control domain. By Ekeland’s variational principle and some basic estimates for state processes and adjoint processes, we establish the necessary conditions for anyε-near optimal control in a local form with an error order of exactε1/2. Moreover, under additional convexity conditions on Hamiltonian function, we prove that anε-maximum condition in terms of the Hamiltonian in the integral form is sufficient for near-optimality of orderε1/2.

Optimal Control Problems of Fully Coupled FBSDEs and Viscosity Solutions of Hamilton--Jacobi--Bellman Equations

In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions to the associated generalized Hamilton--Jacobi--Bellman (HJB) equations. For this we generalize the notion of stochastic backward semigroup introduced by Peng Topics on Stochastic Analysis, Science Press, Beijing, 1997, pp. 85--138. We emphasize that when $\sigma$ depends on the second component of the solution $(Y, Z)$ of the BSDE it makes the stochastic control much more complicated and has as a consequence that the associated HJB equation is combined with an algebraic equation. We prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove a new local existence...

Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control

Existence and uniqueness results of fully coupled forward-backward stochastic differential equations with an arbitrarily large time duration are obtained. Some stochastic Hamilton systems arising in stochastic optimal control systems and mathematical finance can be treated within our framework.