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In this article, we consider the solvability problems for the fully coupled forward-backward stochastic difference equations (FBSΔEs). On one hand, we provide the necessary and sufficient condition for the solvability of the linear FBSΔEs. On the other hand, under the assumption that the coefficients satisfy the monotonicity condition, we investigate the existence and uniqueness theorems for the general non linear FBSΔEs. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE. [ABSTRACT FROM AUTHOR]

In this paper, we propose a general modeling framework for optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs that stem from rational expectations models. We propose a new decoupling technique to obtain the optimal feedback control, which is accompanied by one kind of non-Riccati-type ordinary differential equation (ODE). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs. The obtained results make it possible to compute the control and value function. For this FBLQ problem, the optimal control should depend on the entire trajectory of the state process. Several examples are given to illustrate our results. Funding: M. Hu's research was supported by the National Science Foundation (NSF) [Grant 11671231] and the Young Scholars Program of Shandong University [Grant 2016WLJH10]. S. Ji's research was supported by the NSF [Grant 11571203]. X. Xue's research was supported by "The Fundamental Research Funds of Shandong University," the NSF [Grants 12001316 and 61907022], and the Natural Science Foundation of Shandong Province [Grant ZR2019BF015]. [ABSTRACT FROM AUTHOR]

We design a multi-step predictor-corrector scheme for backward stochastic differential equations (BSDEs). This scheme tries its best to retain the simplicity and improve its convergence rate as much as possible. We investigate the stability and rigorously deduce the error estimates of this scheme. Numerical experiments are compared with the scheme given by Gobet et al. (Math Comput, 85(299): 1359-1391, 2016a) and are given to illustrate that the multi-step predictor-corrector scheme is an efficient probabilistic numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability problems for the fully coupled forward–backward stochastic difference equations (FBSΔEs) on spaces related to discrete time, finite state processes. On one hand, we provide the necessary and sufficient condition for the solvability of the linear FBSΔEs. On the other hand, under the assumption that the coefficients satisfy the monotonicity condition, we investigate the existence and uniqueness theorems for the general nonlinear FBSΔEs. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under G-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution to a type of fully nonlinear HJB equation. [ABSTRACT FROM AUTHOR]

Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim.56 (2018) 4309–4335] and the dynamic programming principle (DPP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim.57 (2019) 3911–3938] for a fully coupled forward–backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton–Jacobi–Bellman (HJB) equation combine an algebra equation respectively. With the help of a new decoupling technique, we obtain the desirable estimates for the fully coupled forward–backward variational equations and establish the relationship. Furthermore, for the smooth case, we discover the connection between the derivatives of the solution to the algebra equation and some terms in the first-order and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as from J. Li and Q. Wei [SIAM J. Control Optim.52 (2014) 1622–1662] and Z. Wu [Syst. Sci. Math. Sci.11 (1998) 249–259], and obtain the relationship between the MP from Z. Wu [Syst. Sci. Math. Sci.11 (1998) 249–259] and the DPP from J. Li and Q. Wei [SIAM J. Control Optim.52 (2014) 1622–1662]. [ABSTRACT FROM AUTHOR]

This paper considers the stochastic linear quadratic optimal control problem in which the control domain is nonconvex. By the functional analysis and convex perturbation methods, we establish a novel maximum principle. The application of the proposed maximum principle is illustrated through a work-out example. [ABSTRACT FROM AUTHOR]

For backward stochastic differential equations (BSDEs), we construct a fully explicit multi-step time-discretization scheme and prove its stability and convergence rates. To approximate conditional expectations in our scheme, we design a new algorithm based on the multilevel Monte Carlo (MLMC) method. This estimator can achieve a prescribed error ϵ with a computational effort of order ϵ − 2. Numerical experiments are given to illustrate the theoretical results of the proposed methods. [ABSTRACT FROM AUTHOR]

We define fully coupled forward-backward stochastic differential equations on spaces related to continuous-time finite-state Markov chains. Existence and uniqueness results of the fully coupled forward-backward stochastic differential equations on Markov chains are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by $$G$$ -Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of $$G$$ -expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the backward stochastic differential equations driven by a -Brownian motion in the following form: where is a decreasing -martingale. Under Lipschitz conditions of and in and , the existence and uniqueness of the solution of the above BSDE in the -framework is proved. [Copyright &y& Elsevier]

Abstract: We study a stochastic optimal control problem where the controlled system is described by a fully coupled forward–backward stochastic differential equation (FBSDE), while the forward state is constrained in a convex set at the terminal time. By introducing an equivalent backward control problem, we use terminal variation approach to obtain a stochastic maximum principle. Applications to the utility optimization problem in the financial market and state constrained stochastic linear quadratic control models are investigated. [Copyright &y& Elsevier]