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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 aim to solve the high dimensional stochastic optimal control problem from the view of the stochastic maximum principle via deep learning. By introducing the extended Hamiltonian system which is essentially a Forward Backward Stochastic Differential Equation (FBSDE) with a maximum condition, we reformulate the original control problem as a new one. According to whether the optimal control has an explicit representation, three algorithms are proposed to solve the new control problem. Numerical results for different examples demonstrate the effectiveness of our proposed algorithms, especially in high dimensional cases. And even if the optimal control u ~ in the maximum condition may not be solved explicitly, our algorithms can still deal with the stochastic optimal control problem. An important application of our proposed method is to calculate the sub-linear expectations, which correspond to a kind of fully nonlinear Partial Differential Equations (PDEs). [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]

This paper solves an optimal insurance design problem in which both the insurer and the insured are subject to Knightian uncertainty about the loss distribution. The Knightian uncertainty is modeled in a multi-prior g-expectation framework. We obtain an endogenous characterization of the optimal indemnity that extends classical theorems of Arrow (Essays in the Theory of Risk Bearing. Markham, Chicago ) and Raviv (Am Econ Rev 69(1):84-96, ) in the classical situation. In the presence of Knightian uncertainty, it is shown that the optimal insurance contract is not only contingent on the realized loss but also on another source of uncertainty coming from the ambiguity. [ABSTRACT FROM AUTHOR]