Forward Backward Stochastic Differential Research Articles

A Class of Efficient Multistep Methods for Forward Backward Stochastic Differential Equations

A Class of Efficient Multistep Methods for Forward Backward Stochastic Differential Equations

Hybrid Neural Networks for Solving Fully Coupled, High-Dimensional Forward–Backward Stochastic Differential Equations

The theory of forward–backward stochastic differential equations occupies an important position in stochastic analysis and practical applications. However, the numerical solution of forward–backward stochastic differential equations, especially for high-dimensional cases, has stagnated. The development of deep learning provides ideas for its high-dimensional solution. In this paper, our focus lies on the fully coupled forward–backward stochastic differential equation. We design a neural network structure tailored to the characteristics of the equation and develop a hybrid BiGRU model for solving it. We introduce the time dimension based on the sequence nature after discretizing the FBSDE. By considering the interactions between preceding and succeeding time steps, we construct the BiGRU hybrid model. This enables us to effectively capture both long- and short-term dependencies, thus mitigating issues such as gradient vanishing and explosion. Residual learning is introduced within the neural network at each time step; the structure of the loss function is adjusted according to the properties of the equation. The model established above can effectively solve fully coupled forward–backward stochastic differential equations, effectively avoiding the effects of dimensional catastrophe, gradient vanishing, and gradient explosion problems, with higher accuracy, stronger stability, and stronger model interpretability.

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.

Learning High-Dimensional McKean–Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence

Learning High-Dimensional McKean–Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence

Stochastic linear quadratic optimal control problem with terminal state constraints

AbstractThis paper addresses the stochastic linear quadratic (LQ) control problem with first‐ and second‐order moment constraints on the terminal state. The problem is a modified version of the optimal covariance control problem, where the terminal state is steered to a given probability distribution. Studying a multiplicative‐noise stochastic system rather than an additive‐noise system is a salient feature. Unlike the existing ideas in the optimal steering, by using the Lagrange multipliers method and establishing the stochastic maximum principle, our problem is converted into solving forward–backward stochastic difference equations (FBSDEs), which is a special stochastic two‐point boundary‐value problem (TPBVP). We provide the optimal closed‐loop controller and necessary and sufficient solvability conditions by developing a nonhomogeneous relationship between the optimal state and costate in FBSDEs. Finally, numerical examples are given to demonstrate our results.

A Strong Convergence Rate of the Averaging Principle for Two-Time-Scale Forward-Backward Stochastic Differential Equations

A Strong Convergence Rate of the Averaging Principle for Two-Time-Scale Forward-Backward Stochastic Differential Equations

Solvability of one kind of forward-backward stochastic difference equations

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.

Backward Stackelberg Games with Delay and Related Forward–Backward Stochastic Differential Equations

In this paper, we study a kind of Stackelberg game where the controlled systems are described by backward stochastic differential delayed equations (BSDDEs). By introducing a new kind of adjoint equation, we establish the sufficient verification theorem for the optimal strategies of the leader and the follower in a general case. Then, we focus on the linear–quadratic (LQ) backward Stackelberg game with delay. The backward Stackelberg equilibrium is presented by the generalized fully coupled anticipated forward–backward stochastic differential delayed Equation (AFBSDDE), which is composed of anticipated stochastic differential equations (ASDEs) and BSDDEs. Moreover, we obtain the unique solvability of the AFBSDDE using the continuation method. As an application of the theoretical results, the pension fund problem with delay effect is considered.

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

The Kolmogorov infinite dimensional equation in a Hilbert space via deep learning methods

We consider the nonlinear Kolmogorov equation posed in a Hilbert space H, not necessarily of finite dimension. This model was recently studied by Cox et al. [12] in the framework of weak convergence rates of stochastic wave models. Here, we propose a complementary approach by providing an infinite-dimensional Deep Learning method to approximate suitable solutions of this model. Based in the work by Hure, Pham and Warin [25] concerning the finite dimensional case, and our previous work [9] dealing with Lévy based processes, we generalize an Euler scheme and consistency results for the Forward Backward Stochastic Differential Equations to the infinite dimensional Hilbert valued case. Since our framework is general, we require the recently developed DeepOnets neural networks [10,29] to describe in detail the approximation procedure. Also, the framework developed by Fuhrman and Tessitore [17] to fully describe the stochastic approximations will be adapted to our setting.

Propagation of Chaos of Forward–Backward Stochastic Differential Equations with Graphon Interactions

In this paper, we study graphon mean field games using a system of forward–backward stochastic differential equations. We establish the existence and uniqueness of solutions under two different assumptions and prove the stability with respect to the interacting graphons which are necessary to show propagation of chaos results. As an application of propagation of chaos, we prove the convergence of n-player game Nash equilibrium for a general model, which is new in the theory of graphon mean field games.

Backward Deep BSDE Methods and Applications to Nonlinear Problems

We present a pathwise deep Backward Stochastic Differential Equation (BSDE) method for Forward Backward Stochastic Differential Equations with terminal conditions that time-steps the BSDE backwards and apply it to the differential rates problem as a prototypical nonlinear problem of independent financial interest. The nonlinear equation for the backward time-step is solved exactly or by a Taylor-based approximation. This is the first application of such a pathwise backward time-stepping deep BSDE approach for problems with nonlinear generators. We extend the method to the case when the initial value of the forward components X can be a parameter rather than fixed and similarly to also learn values at intermediate times. We present numerical results for a call combination and for a straddle, the latter comparing well to those obtained by Forsyth and Labahn with a specialized PDE solver.

Strong Stability Preserving Multistep Schemes for Forward Backward Stochastic Differential Equations

Strong Stability Preserving Multistep Schemes for Forward Backward Stochastic Differential Equations

Solution to Forward–Backward Stochastic Differential Equations With Random Coefficients and Application to Deterministic Optimal Control

We address forward–backward stochastic differential equations (FBSDEs) with random coefficients. Differently from previous approaches, we consider FBSDEs with coefficients that are random and correlated with forward and backward variables. In addition, expectation terms appear in the forward equation, and this makes challenging to solve coupled FBSDEs, e.g., those coming from corporations’ fiscal policy planning. Using a discretization approach, an analytic solution to the FBSDEs is given in terms of coupled stochastic Riccati-type differential equations. We also apply our results to optimal deterministic control problem for Itô stochastic systems with both random and deterministic coefficients. Our results show that the discretization approach is a powerful tool to decouple complex FBSDEs.

Solving Stochastic Optimal Control Problem via Stochastic Maximum Principle with Deep Learning Method

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 \(\tilde{u}\) 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).

Numerical Resolution of McKean-Vlasov FBSDEs Using Neural Networks

We propose several algorithms to solve McKean-Vlasov Forward Backward Stochastic Differential Equations (FBSDEs). Our schemes rely on the approximating power of neural networks to estimate the solution or its gradient through minimization problems. As a consequence, we obtain methods able to tackle both mean-field games and mean-field control problems in moderate dimension. We analyze the numerical behavior of our algorithms on several multidimensional examples including non linear quadratic models.

On Forward–Backward Stochastic Differential Equations in a Domination-Monotonicity Framework

On Forward–Backward Stochastic Differential Equations in a Domination-Monotonicity Framework

Extended Mckean-Vlasov optimal stochastic control applied to smart grid management,

We study the mathematical modeling of the energy management system of a smart grid, related to a aggregated consumer equipped with renewable energy production (PV panels e.g.), storage facilities (batteries), and connected to the electrical public grid. He controls the use of the storage facilities in order to diminish the random fluctuations of his residual load on the public grid, so that intermittent renewable energy is better used leading globally to a much greener carbon footprint. The optimization problem is described in terms of an extended McKean-Vlasov stochastic control problem. Using the Pontryagin principle, we characterize the optimal storage control as solution of a certain McKean-Vlasov Forward Backward Stochastic Differential Equation (possibly with jumps), for which we prove existence and uniqueness. Quasi-explicit solutions are derived when the cost functions may not be linear-quadratic, using a perturbation approach. Numerical experiments support the study.

Two Equivalent Families of Linear Fully Coupled Forward Backward Stochastic Differential Equations

In this paper, we investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDEs) and its applications to optimal Linear Quadratic (LQ) problems. Within these families, one could get same well-posedness of FBSDEs with totally different coefficients. A family of FBSDEs is proved to be equivalent with respect to the Unified Approach. Thus one could get well-posedness of whole family once a member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. Owing to the coupling structure between forward and backward equations, it leads to a highly interdependence in solutions. We are able to decouple FBSDEs into partial coupling, by virtue of linear transformation, without losing the existence and uniqueness to solutions. Moreover, owing to non-degeneracy of transformation matrix, the solution to original FBSDEs is totally determined by solutions of FBSDEs after transformation. In addition, an example of optimal LQ problem is presented to illustrate.

Solvability of forward–backward stochastic difference equations with finite states

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.