Forward-backward Stochastic Differential Equations Research Articles

Approximation of a degenerate semilinear PDE with a nonlinear Neumann boundary condition

We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform ellipticity is not required for the diffusion coefficient. We show that this problem admits a viscosity solution which can be approximated by a penalization. The Lipschitz condition is required for the coefficients of the diffusion part. The nonlinear part as well as the Neumann condition are Lipschitz. Moreover, the nonlinear part is monotone in the solution variable. Note that the existence of a viscosity solution to this problem has been established in [13] then completed in [15]. In the present paper, we construct a sequence of penalized systems of decoupled forward backward stochastic differential equations (FBSDEs) then we directly show its strong convergence. This allows us to deal with the case where the nonlinearity depends on both the solution and its gradient. Our work extends, in particular, the result of [4] and, in some sense, those of [1, 3]. In contrast to works [1, 3, 4], we do not pass by the weak compactness of the laws of the stochastic system associated to our problem.

Pricing barrier options with deep backward stochastic differential equation methods

This paper presents a novel and direct approach to solving boundary- and final-value problems, corresponding to barrier options, using forward pathwise deep learning and forward–backward stochastic differential equations (FBSDEs). Barrier instruments are instruments that expire or transform into another instrument if a barrier condition is satisfied before maturity; otherwise they perform like the instrument without the barrier condition. In a partial differential equation, this corresponds to adding boundary conditions to the final-value problem. The deep backward stochastic differential equation (deep BSDE) methods developed so far have not addressed barrier/boundary conditions directly. We extend the pathwise deep BSDE methods to the barrier condition case by adding nodes to the computational graph, in order to explicitly monitor the barrier conditions for each realization of the dynamics, as well as adding nodes that preserve the time, state variables and trading strategy value at the barrier breach, or at maturity otherwise. Given these additional nodes in the computational graph, the forward loss function quantifies the replication of the barrier or final payoff according to a chosen risk measure such as the squared sum of differences. The proposed method can handle any barrier condition in the FBSDE setup and any Dirichlet boundary conditions in the partial differential equation setup, in both low and high dimensions.

A stochastic linear-quadratic differential game with time-inconsistency

<abstract><p>We consider a general stochastic linear-quadratic differential game with time-inconsistency. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derive a sufficient condition for equilibrium strategies via a system of forward-backward stochastic differential equation. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given.</p></abstract>

Dynamic optimization problems for mean-field stochastic large-population systems

This paper considers dynamic optimization problems for a class of control average meanfield stochastic large-population systems. For each agent, the state system is governed by a linear mean-field stochastic differential equation with individual noise and common noise, and the weight coefficients in the corresponding cost functional can be indefinite. The decentralized optimal strategies are characterized by stochastic Hamiltonian system, which turns out to be an algebra equation and a mean-field forward-backward stochastic differential equation. Applying the decoupling method, the feedback representation of decentralized optimal strategies is further obtained through two Riccati equations. The solvability of stochastic Hamiltonian system and Riccati equations under indefinite condition is also derived. The explicit structure of the control average limit and the related mean-field Nash certainty equivalence equation systems are discussed by some separation techniques. Moreover, the decentralized optimal strategies are proved to satisfy the approximate Nash equilibrium property. The good performance of the proposed theoretical results is illustrated by a practical example from the engineering field.

Extended mean-field control problem with partial observation

We study an extended mean-field control problem with partial observation, where the dynamic of the state is given by a forward-backward stochastic differential equation of McKean-Vlasov type. The cost functional, the state and the observation all depend on the joint distribution of the state and the control process. Our problem is motivated by the recent popular subject of mean-field games and related control problems of McKean-Vlasov type. We first establish a necessary condition in the form of Pontryagin’s maximum principle for optimality. Then a verification theorem is obtained for optimal control under some convex conditions of the Hamiltonian function. The results are also applied to studying linear-quadratic mean-filed control problem in the type of scalar interaction.

Mean-field type FBSDEs in a domination-monotonicity framework and LQ multi-level Stackelberg games

<p style='text-indent:20px;'>Motivated by various mean-field type linear-quadratic (MF-LQ, for short) multi-level Stackelberg games, we propose a kind of multi-level self-similar randomized domination-monotonicity structures. When the coefficients of a class of mean-field type forward-backward stochastic differential equations (MF-FBSDEs, for short) satisfy this kind of structures, we prove the existence, the uniqueness, an estimate and the continuous dependence on the coefficients of solutions. Further, the theoretical results are applied to construct unique Stackelberg equilibria for forward and backward MF-LQ multi-level Stackelberg games, respectively.</p>

General indefinite backward stochastic linear-quadratic optimal control problems

A general backward stochastic linear-quadratic optimal control problem is studied, in which both the state equation and the cost functional contain the nonhomogeneous terms. The main feature of the problem is that the weighting matrices in the cost functional are allowed to be indefinite, and the cross-product terms in the control and the state processes are present. Necessary and sufficient conditions for the solvability of the problem are obtained, and a characterization of the optimal control in terms of forward-backward stochastic differential equations is derived. By a Riccati equation approach, a general procedure for constructing optimal controls is developed and the value function is obtained explicitly.

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.

Newton-Kantorovitch method for decoupled forward-backward stochastic differential equations

We formulate a Newton-Kantorovitch method for solving decoupled forward-backward stochastic differential equations involving smooth and degenerate coefficients with uniformly bounded derivatives. We show that it converges globally and its rate of convergence is exponential.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/98/abstr.html

An averaging principle for nonlinear parabolic PDEs via FBSDEs driven by G-Brownian motion

In this paper, we study the averaging problem for a class of forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) with rapidly oscillating coefficients, which corresponds to the singular perturbation problem of a kind of fully nonlinear partial differential equations (PDEs). With the help of the nonlinear stochastic analysis techniques and viscosity solution methods, we prove that the limit distribution of the solution is the unique viscosity solution to a fully nonlinear PDE.

A linear-quadratic partially observed Stackelberg stochastic differential game with application

This paper is concerned with a linear-quadratic partially observed Stackelberg stochastic differential game with correlated state and observation noises, where the control set is not necessarily convex. Both the leader and the follower have their own observation equations, and the information filtration available to the leader is contained in that available to the follower. Necessary and sufficient conditions of the Stackelberg equilibrium points are derived. In the follower’s problem, the state estimation feedback of optimal control can be represented by a forward-backward stochastic differential filtering equation and some Riccati equation. In the leader’s problem, via the innovation process, the state estimation feedback of optimal control is represented by a stochastic differential filtering equation, a semi-martingale process and three high-dimensional Riccati equations. As an application, a dynamic advertising problem with asymmetric information is studied, and the effectiveness and reasonability of the theoretical result are illustrated by numerical simulations.

Sup-Inf/Inf-Sup Problem on Choice of a Probability Measure by Forward–Backward Stochastic Differential Equation Approach

This article presents a problem on model uncertainties in stochastic control, in which an agent assumes a best case scenario on one risk and at the same time a worst case scenario on another risk. Particularly, the agent maximizes its view on a Brownian motion, simultaneously minimizing its view on another Brownian motion in choice of a probability measure. This selection method of a probability measure generalizes an approach to model uncertainties in which one considers the worst case scenarios for the views on Brownian motions, such as in the robust control. Specifically, we newly formulate and solve this problem based on a backward stochastic differential equation (BSDE) approach as a sup-inf (resp., inf-sup) optimal control problem on choice of a probability measure with the control domains dependent on stochastic processes. Concretely, we show that under certain conditions, the sup-inf and inf-sup problems are equivalent and these are solved by finding a solution of a BSDE with a stochastic Lipschitz driver. Then, we investigate two cases in which the optimal probability measure is explicitly obtained. The expression of the optimal probability measure includes signs of the diffusion terms of the value process, which are hard to determine in general. In these cases, we show two methods of determining the signs:the first one is by comparison theorems, and the second one is to predetermine the signs a priori and confirm them afterward by explicitly solving the corresponding equations.

Numerical approximations of coupled forward–backward SPDEs

We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward–backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. Precisely, we generalize the pioneering work of Dunst and Prohl [SIAM J. Sci. Comp., 38(2017), 2725–2755] by considering general nonlinear and nonlocal FBSPDEs with more inclusive coupling; self-contained proofs are provided and different numerical techniques for the resulting finite dimensional equations are adopted. For such FBSPDEs allowing for non-Lipschitz nonlinearity on unknown variables, we prove the existence and uniqueness of the strong solution as well as of the weak solution. Then, the finite element method in the spatial domain leads to approximations of FBSPDEs by finite-dimensional forward–backward stochastic differential equations which are numerically computed by using some deep learning-based schemes. A strong convergence analysis is addressed for the spatial discretization of coupled FBSPDEs under an additional Lipschitz assumption, and the numerical examples, including both decoupled and coupled cases, indicate that our methods are quite efficient.

LQ control of forward and backward stochastic difference system

This paper studies linear quadratic (LQ) control of forward and backward stochastic difference system (FBSDS). There are two main contributions: first, the equivalent condition is obtained for the solvability of optimal control of FBSDS with multi-multiplicative noises under complete information and the analytic solution is also stated by the proposed Riccati equation. Second, the result of complete information is deduced to the situation of incomplete information, that is, the equivalent conditions of solvability and analytical solution are obtained based on Riccati equation. The main technique we use is to solve the forward–backward stochastic difference equations (FBSDEs) originated from maximum principle (MP).

Strong solutions of forward–backward stochastic differential equations with measurable coefficients

This paper investigates solvability of fully coupled systems of forward–backward stochastic differential equations (FBSDEs) with irregular coefficients. In particular, we assume that the coefficients of the FBSDEs are merely measurable and bounded in the forward process. We crucially use compactness results from the theory of Malliavin calculus to construct strong solutions. Despite the irregularity of the coefficients, the solutions turn out to be differentiable, at least in the Malliavin sense and, as functions of the initial variable, in the Sobolev sense.

A BSDE Approach to Stochastic Differential Games Involving Impulse Controls and HJBI Equation

This paper focuses on zero-sum stochastic differential games in the framework of forward-backward stochastic differential equations on a finite time horizon with both players adopting impulse controls. By means of BSDE methods, in particular that of the notion from Peng’s stochastic backward semigroups, the authors prove a dynamic programming principle for both the upper and the lower value functions of the game. The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle. As a consequence, the uniqueness implies that the upper and lower value functions coincide and the game admits a value.

Forward-backward stochastic equations: a functional fixed point approach

We introduce forward-backward stochastic equations (FBSEs) that incorporate fully-coupled forward-backward structure into backward stochastic equations (BSEs) introduced in Cheridito and Nam (Ann. Probab. 45(6A):3795–3828, 2017). Such a system generalizes the classical backward stochastic differential equations (BSDEs) and forward-backward stochastic differential equations (FBSDEs) in previous literature. We transform an FBSE into a fixed point equation on the space of random variables and then apply general fixed point theorems to derive the existence and/or uniqueness of a solution. As a result, we obtain novel existence and/or uniqueness results for fully-coupled FBSDEs with functional drivers, which are either Lipschitz or non-Lipschitz.

Optimal Control and Stabilization for Itô Systems with Input Delay

The paper considers the linear quadratic regulation (LQR) and stabilization problems for Ito stochastic systems with two input channels of which one has input delay. The underlying problem actually falls into the field of asymmetric information control because of the nonidentical measurability induced by the input delay. In contrast with single-channel single-delay problems, the challenge of the problems under study lies in the interaction between the two channels which are measurable with respect to different filtrations. The key techniques conquering such difficulty are the stochastic maximum principle and the orthogonal decomposition and reorganization technique proposed in a companion paper. The authors provide a way to solve the delayed forward backward stochastic differential equation (D-FBSDE) arising from the maximum principle. The necessary and sufficient solvability condition and the optimal controller for the LQR problem are given in terms of a new Riccati differential equation established herein. Further, the necessary and sufficient stabilization condition in the mean square sense is provided and the optimal controller is given. The idea proposed in the paper can be extended to solve related control problems for stochastic systems with multiple input channels and multiple delays.

LATENCY AND LIQUIDITY RISK

Latency (i.e. time delay) in electronic markets affects the efficacy of liquidity taking strategies. During the time liquidity, takers process information and send marketable limit orders (MLOs) to the exchange, the limit order book (LOB) might undergo updates, so there is no guarantee that MLOs are filled. We develop a latency-optimal trading strategy that improves the marksmanship of liquidity takers. The interaction between the LOB and MLOs is modeled as a marked point process. Each MLO specifies a price limit so the order can receive worse prices and quantities than those the liquidity taker targets if the updates in the LOB are against the interest of the trader. In our model, the liquidity taker balances the tradeoff between the costs of missing trades and the costs of walking the book. In particular, we show how to build cost-neutral strategies, that on average, trade price improvements for fewer misses. We employ techniques of variational analysis to obtain the price limit of each MLO the agent sends. The price limit of an MLO is characterized as the solution to a class of forward–backward stochastic differential equations (FBSDEs) driven by random measures. We prove the existence and uniqueness of the solution to the FBSDE and numerically solve it to illustrate the performance of the latency-optimal strategies.

Linear-quadratic generalized Stackelberg games with jump-diffusion processes and related forward-backward stochastic differential equations

Linear-quadratic generalized Stackelberg games with jump-diffusion processes and related forward-backward stochastic differential equations