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We study backward stochastic differential equations (BSDEs) for time-changed L\'evy noises when the time-change is independent of the L\'evy process. We prove existence and uniqueness of the solution and we obtain an explicit formula for…
We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type): \[ \left\{\begin{array} [c]{r} dX(t)+\partial\varphi\left(X(t)\right)…
Control of the stochastic dynamics of a quantum system is indispensable in fields such as quantum information processing and metrology. However, there is no general ready-made approach to the design of efficient control strategies. Here, we…
This paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation.…
This paper is concerned with optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs. In order to obtain the state feedback representation of the optimal control, we…
We consider some certain nonlinear perturbations of the stochastic linear-quadratic optimization problems and study the connections between their solutions and the corresponding Markovian backward stochastic diferential equations (BSDEs).…
This paper explores a class of fully coupled nonlinear forward-backward stochastic difference equations (FBS$\Delta$Es). Building on insights from linear quadratic optimal control problems, we introduce a more relaxed framework of…
This script offers an implementation-oriented introduction to deep learning methods for solving and estimating high-dimensional dynamic stochastic models in economics and finance. Its starting point is the curse of dimensionality:…
In this paper we present a dynamic programing approach to stochastic optimal control problems with dynamic, time-consistent risk constraints. Constrained stochastic optimal control problems, which naturally arise when one has to consider…
Linear-quadratic optimal control problem for systems governed by forward-backward stochastic differential equations has been extensively studied over the past three decades. Recent research has revealed that for forward-backward control…
In this paper, we study a kind of optimal control problem for forward-backward stochastic differential equations (FBSDEs for short) of McKean--Vlasov type via the dynamic programming principle (DPP for short) motivated by studying the…
This paper presents a novel approach to numerically solve stochastic differential games for nonlinear systems. The proposed approach relies on the nonlinear Feynman-Kac theorem that establishes a connection between parabolic deterministic…
This paper is devoted to a stochastic differential game of functional forward-backward stochastic differential equation (FBSDE, for short). The associated upper and lower value functions of the stochastic differential game are defined by…
We construct an aggregated version of the value processes associated with stochastic control problems, where the criterion to optimise is given by solutions to semi-martingale backward stochastic differential equations (BSDEs). The results…
Recently, the deep learning method has been used for solving forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). It has good accuracy and performance for high-dimensional…
A modified Deep BSDE (backward differential equation) learning method with measurability loss, called Deep BSDE-ML method, is introduced in this paper to solve a kind of linear decoupled forward-backward stochastic differential equations…
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs),…
One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the…
This paper is concerned with a stochastic linear quadratic (LQ, for short) control problem with a recursive cost functional. It involves BSDEs in $L^1$ whose well-posedness is a subtle issue. A suitable framework has been adopted so that…
The optimal control problem for discrete-time systems with colored multiplicative noise is discussed in this paper. The problem will be more difficult to deal with than the case of white noise due to the correlation of the adjoining state.…