Forward-backward Stochastic Differential Equations Research Articles

Forward-backward stochastic differential equations driven by G-Brownian motion under weakly coupling condition

In this paper, we obtain the existence and uniqueness theorem of Lp-solution for coupled forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) with arbitrary T under weakly coupling condition. Specially, the result for p∈(1,2) is completely different from the one for p≥2. Furthermore, by considering the dual linear FBSDE under a suitable reference probability, we establish the comparison theorem for G-FBSDEs under weakly coupling condition.

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.

Convergence of a Robust Deep FBSDE Method for Stochastic Control

In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems

We prove the local well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter t on top of the temporal and spatial variables (s,y) and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point (t,s,y) but also at the diagonal line of the time domain (s,s,y). Such equations arise from time-inconsistent problems in game theory or behavioral economics, where the observations and preferences are (reference-)time-dependent. We first study the linearized version of the nonlocal PDEs with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. With fixed-point arguments, we obtain the solvability of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we establish the well-posedness under the fully nonlinear case. Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic PDE is an adapted solution to a flow of second-order forward-backward stochastic differential equations.

A linear-quadratic mean-field game of backward stochastic differential equation with partial information and common noise

This paper studies a linear-quadratic (LQ) mean-field game of stochastic large-population system with partial information and common noise, where the large-population system satisfies a class of backward stochastic differential equations (BSDEs), and both coupling structure and ki (a part of solution of BSDE) enter state equations and cost functionals. By virtue of stochastic maximum principle and optimal filter technique, we obtain a Hamiltonian system first, which is a fully-coupled forward-backward stochastic differential equation (FBSDE). Decoupling the Hamiltonian system, we obtain three ordinary differential equations (ODEs), a forward and a backward optimal filtering equations, which enable us to derive an optimal control of an auxiliary limiting control problem. Next, we verify that a decentralized control strategy is an ϵ-Nash equilibrium of the LQ game. Finally, we solve a product pricing problem and a network security control problem in applications. We give a near-optimal price strategy and a near-optimal control strategy with numerical simulations, respectively.

Strong Stability Preserving Multistep Schemes for Forward Backward Stochastic Differential Equations

Strong Stability Preserving Multistep Schemes for Forward Backward Stochastic Differential Equations

Mean-Field Type FBSDEs under Domination-Monotonicity Conditions and Application to LQ Problems

This paper is concerned with a class of mean-field type coupled forward-backward stochastic differential equations (MF-FBSDEs), in which the coupling appears in integral terms, terminal terms, and initial terms. Inspired by various mean-field type linear-quadratic (MF-LQ) optimal control problems, we propose a type of randomized domination-monotonicity conditions, under which and the usual Lipschitz condition we obtain a well-posedness result on MF-FBSDEs in the sense of square integrability including the unique solvability, an estimate of the solution, and the related continuous dependence property of the solution on the coefficients. The result of MF-FBSDEs in turn extends MF-LQ problems in the literature to a general situation where the initial states or the terminal states are also controlled at the same time, and gives explicit expressions of the related unique optimal controls.

Numerical schemes for fully coupled mean-field forward backward stochastic differential equations

In this paper, based on the nonlinear Feynman-Kac formula and Itô-Taylor schemes for solving mean-field forward stochastic differential equations, we propose numerical schemes for solving fully coupled mean-field forward backward stochastic differential equations. Our numerical experiments show that the schemes are stable, effective, and can be of second-order accuracy when weak second-order schemes are used for solving mean-field forward stochastic differential equations.

Approximation and error analysis of forward–backward SDEs driven by general Lévy processes using shot noise series representations

We consider the simulation of a system of decoupled forward–backward stochastic differential equations (FBSDEs) driven by a pure jump Lévy process L and an independent Brownian motion B. We allow the Lévy process L to have an infinite jump activity. Therefore, it is necessary for the simulation to employ a finite approximation of its Lévy measure. We use the generalized shot noise series representation method by [26] to approximate the driving Lévy process L. We compute the Lp error, p ≥ 2, between the true and the approximated FBSDEs which arises from the finite truncation of the shot noise series (given sufficient conditions for existence and uniqueness of the FBSDE). We also derive the Lp error between the true solution and the discretization of the approximated FBSDE using an appropriate backward Euler scheme.

Forward-backward stochastic differential equations: Initiation, development and beyond

<p style='text-indent:20px;'>Jin Ma has made fundamental contributions to the theory of forward-backward stochastic differential equations (FBSDEs, for short). In this paper, as one of his main collaborators, I will recall the initiation, the path of the development for FBSDEs, as well as some interesting historic stories.</p>

Linear-Quadratic-Gaussian mean-field controls of social optima

<p style='text-indent:20px;'>This paper investigates a class of unified stochastic linear-quadratic-Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a generalized setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be <i>indefinite</i> in cost functional. This setup is innovative and has great theoretical and realistic significance as its applications in mathematical finance (e.g., portfolio selection in mean-variation model). Using some <i>fully-coupled</i> variational analysis under the person-by-person optimality principle, and the mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with <i>embedding representation</i>. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.</p>

Sobolev space weak solutions to one kind of quasilinear parabolic partial differential equations related to forward-backward stochastic differential equations

This paper is concerned with the Sobolev type weak solutions of one class of second order quasilinear parabolic partial differential equations (PDEs, for short). First of all, similar to Feng, Wang and Zhao [9] and Wu and Yu [29], we use a family of coupled forward-backward stochastic differential equations (FBSDEs, for short) which satisfy the monotonous assumption to represent the classical solutions of the quasilinear PDEs. Then, based on the classical solutions of a family of PDEs approximating the weak solutions of the quasilinear PDEs, we prove the existence of the weak solutions. Moreover, the principle of norm equivalence is employed to link FBSDEs and PDEs to obtain the uniqueness of the weak solutions. In summary, we provide a probabilistic interpretation for the weak solutions of quasilinear PDEs, which enriches the theory of nonlinear PDEs.

On path-dependent multidimensional forward-backward SDEs

<p style='text-indent:20px;'>This paper extends the results of Ma, Wu, Zhang, Zhang [<xref ref-type="bibr" rid="b11">11</xref>] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> can depend on the whole path of the forward process up to time <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling field on small time interval. Then by introducing the characteristic BSDE, we show that a global decoupling field can be constructed by patching local solutions together as long as the solution of the characteristic BSDE remains bounded. Finally, we provide a stability result for path-dependent forward-backward SDEs.</p>

Deep signature FBSDE algorithm

<p style='text-indent:20px;'>We propose a deep signature/log-signature FBSDE algorithm to solve forward-backward stochastic differential equations (FBSDEs) with state and path dependent features. By incorporating the deep signature/log-signature transformation into the recurrent neural network (RNN) model, our algorithm shortens the training time, improves the accuracy, and extends the time horizon comparing to methods in the existing literature. Moreover, our algorithms can be applied to a wide range of applications such as state and path dependent option pricing involving high-frequency data, model ambiguity, and stochastic games, which are linked to parabolic partial differential equations (PDEs), and path-dependent PDEs (PPDEs). Lastly, we also derive the convergence analysis of the deep signature/log-signature FBSDE algorithm.</p>

Global existence for quadratic FBSDE systems and application to stochastic differential games

In this note, we extend some recent results on systems of backward stochastic differential equations (BSDEs) with quadratic growth to the case of coupled forward-backward stochastic differential equations (FBSDEs). We work in a Markovian setting, and use results from the quadratic BSDE literature together with PDE techniques to obtain a-priori estimates which lead to an existence result. We also identify a general class of stochastic differential games whose corresponding FBSDE systems are covered by our main existence result. This leads to the existence of Markovian Nash equilibria for such games.

Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems

<p style='text-indent:20px;'>This paper studies open-loop equilibriums for a general class of time-inconsistent stochastic control problems under jump-diffusion SDEs with deterministic coefficients. Inspired by the idea of Four-Step-Scheme for forward-backward stochastic differential equations with jumps (FBSDEJs, for short), we derive two systems of integro-partial differential equations (IPDEs, for short). Then, we rigorously prove a verification theorem which provides a sufficient condition for open-loop equilibrium strategies. As special cases, a mean-variance portfolio selection problem and a time-inconsistent problem under non-exponential discounting are discussed.</p>

Forward–backward stochastic differential equations with delay generators

In this paper, we prove a result of existence and uniqueness of solutions to coupled forward–backward stochastic differential equations with delayed generators under a Lipschitz condition.

Leader–follower mean field LQ games: A direct method

AbstractThis paper investigates a linear quadratic mean field game with a leader and a large number of followers. The leader first gives its strategy, and then each follower optimizes its own cost. We first solve a mean field game problem, which gives the best response of followers to the leader's strategy. After applying the followers' strategies, the leader is faced to an optimal control problem driven by a high‐dimensional forward‐backward stochastic differential equations (FBSDEs). By decoupling the high‐dimensional Hamiltonian system with mean field approximations, we construct a set of decentralized strategies for all the players, which are further shown to be an ‐Stackelberg equilibrium. ‐

Decentralized strategies for finite population linear–quadratic–Gaussian games and teams

This paper is concerned with a new class of mean-field games which involve a finite number of agents. Necessary and sufficient conditions are obtained for the existence of the decentralized open-loop Nash equilibrium in terms of non-standard forward–backward stochastic differential equations (FBSDEs). By solving the FBSDEs, we design a set of decentralized strategies by virtue of two differential Riccati equations. Instead of the ɛ-Nash equilibrium in classical mean-field games, the set of decentralized strategies is shown to be a Nash equilibrium. For the infinite-horizon problem, a simple condition is given for the solvability of the algebraic Riccati equation arising from consensus. Furthermore, the social optimal control problem is studied. Under a mild condition, the decentralized social optimal control and the corresponding social cost are given.

A sufficient condition for optimal control problem of fully coupled forward‐backward stochastic systems with jumps: A state‐constrained control approach

AbstractWe study the stochastic optimal control problem for fully coupled forward‐backward stochastic differential equations (FBSDEs) with jump diffusions. A major technical challenge of such problems arises from the dependence of the (forward) diffusion term on the backward SDE and the presence of jump diffusions. Previously, this class of problems has been solved via only the stochastic maximum principle, which guarantees only the necessary condition of optimality and requires identifying unknown parameters in the corresponding variational inequality. Our paper provides an alternative approach, which constitutes the sufficient condition for optimality. Specifically, the original fully coupled FBSDE control problem (referred to as (P)) is converted into the terminal state‐constrained forward stochastic control problem (referred to as ) that includes additional (possibly unbounded) control variables. Then is solved via the backward reachability analysis, by which the value function of is expressed as the zero‐level set of the value function for the auxiliary unconstrained (forward) control problem (referred to as ). Unlike ), is an unconstrained problem, which includes additional control variables as a consequence of the martingale representation theorem. We show that the value function for is the unique viscosity solution to the associated integro‐type Hamilton‐Jacobi‐Bellman (HJB) equation. The viscosity solution analysis presented in our paper requires a new technique due to additional control variables in the Hamiltonian maximization and the presence of the nonlocal integral operator in terms of the (singular) Lévy measure. To solve the original problem (P), we reverse our approach. Specifically, we first solve to obtain the value function using the verification theorem and the viscosity solution of the HJB equation. Then is solved by characterizing the zero‐level set of the value function of , from which the optimal solution of (P) can be constructed. To illustrate the theoretical results of this paper, applications to the linear‐quadratic problem for fully coupled FBSDEs with jumps are also presented.