Linear Backward Stochastic Differential Equations Research Articles

Robust backward linear-quadratic differential game and team: A soft-constraint analysis

The robust design with model uncertainty is an extensively investigated research issue but traditionally focuses on forward stochastic system for which the initial condition is given as a priori. This paper makes a systematic continuation of this issue but in backward linear-quadratic context. That is, the state dynamics is evolved by a linear backward stochastic differential equation with terminal condition being specified, and cost functionals to be optimized are of quadratic forms. Three types of dynamic backward decisions, are introduced in both game and team setting in presence of drift uncertainty (disturbance). Specifically, in (Stackelberg) game setting, both cases for the follower or leader with unknown disturbance are discussed respectively; whereas in the team setting, the case with disturbance unknown to the leader is highlighted. For each of these three types, corresponding robust backward strategies are designed and associated optimal values for either leader or follower are given.

Optimal regulator for a class of nonlinear stochastic systems with random coefficients

We consider an optimal regulator problem for a class of nonlinear stochastic systems with a square-root nonlinearity and random coefficients, and using the quadratic-linear criterion. This represents a certain nonlinear generalisation of the stochastic linear-quadratic control problem with random coefficients. The solution if found in an explicit closed-form as an affine state-feedback control in terms of a Riccati and linear backward stochastic differential equations. As an application, we give the solution to an optimal investment problem in a market with random coefficients.

Stochastic linear-quadratic control with a jump and regime switching on a random horizon

In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a random horizon $ [0,T\wedge\tau] $, where $ \tau $ is a given random jump time for the underlying state process and $ T $ a constant. We obtain the explicit optimal feedback control and explicit optimal value of the problem by solving a system of stochastic Riccati equations (SREs) with jumps on the random horizon $ [0,T\wedge\tau] $. By the decomposition approach stemming from filtration enlargement theory, we express the solution to the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon $ [0,T] $. Solving the latter system is the key theoretical contribution of this paper and we accomplish this under three different conditions, one of which seems to be new in the literature. The above results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.

Mean-Variance Asset-Liability Management in a Non-Markovian Regime-Switching Jump-Diffusion Market with Random Horizon

This paper investigates the problem of mean-variance asset-liability management (ALM) in a market where the dynamics of assets are non-Markovian regime-switching models driven by a Brownian motion, a Poisson random measure and a continuous time finite-state Markov chain. It is assumed that the time horizon is uncertain relying not only on asset prices and liability values, but also on other factors. The insurer aims to minimize the variance of the terminal surplus given an expected terminal surplus subject to the risk of paying out random liabilities of an extensive Cramer-Lundberg model. By further developing the solvability of a linear backward stochastic differential equation (BSDE) with two types of jumps, namely jumps modelled by a Poisson random measure and by basic martingales related to a Markov chain, closed-form expressions for both the efficient strategy and the efficient frontier are obtained and represented by unique solutions to several relevant BSDEs.

Linear quadratic control of backward stochastic differential equation with partial information

In this paper, we study an optimal control problem of linear backward stochastic differential equation (BSDE) with quadratic cost functional under partial information. This problem is solved completely and explicitly by using a stochastic maximum principle and a decoupling technique. In terms of the maximum principle, a stochastic Hamiltonian system, which is a forward-backward stochastic differential equation (FBSDE) with filtering, is obtained. By decoupling the stochastic Hamiltonian system, three Riccati equations, a BSDE with filtering, and a stochastic differential equation (SDE) with filtering are derived. We then get a feedback representation of optimal control. An explicit formula for the corresponding optimal cost is also established. As illustrative examples, we consider two special scalar-valued control problems and give some numerical simulations.

On backward SPDEs without proper Cauchy condition

We study linear backward stochastic partial differential equations (BSPDEs) of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution over time. We establish well-posedness, existence, uniqueness, and regularity, for the solutions of this new problem. In particular, this can be considered as a possibility to recover a solution of a BSPDE in a setting where its values at the terminal time are unknown, and where the average of the solution over time is preselected.

A Feedback Nash Equilibrium for Affine-Quadratic Zero-Sum Stochastic Differential Games With Random Coefficients

We study the affine-quadratic zero-sum stochastic differential game with random coefficients, where the coefficients of the stochastic differential equation (SDE) are random processes and both additive and state multiplicative noise are included in the diffusion term of the corresponding SDE. By applying Ito-Kunita’s formula to the quadratic random field, we develop a direct approach, also known as the completion of squares method, to characterize the explicit (feedback) Nash equilibrium and obtain the optimal game value. The characterized Nash equilibrium depends linearly on the state and the additional linear backward SDE. We also verify the optimality of the Nash equilibrium by characterizing the smooth solution of the stochastic Hamilton-Jacobi-Isaacs equation that is the second-order stochastic partial differential equation obtained from dynamic programming.

A Simple Proof of Indefinite Linear-Quadratic Stochastic Optimal Control With Random Coefficients

We consider the indefinite linear-quadratic optimal control problem for stochastic systems with random coefficients, where the corresponding cost parameters need not be definite matrices. Although the solution to this problem was obtained previously via the stochastic maximum principle or by solving the stochastic Hamilton–Jacobi–Bellman (HJB) equation, our approach is simple and direct. Specifically, we develop a direct approach, also known as the completion of squares method, to characterize the explicit optimal solution and the optimal cost. The corresponding optimal solution is linear characterized in terms of the stochastic Riccati differential equation (SRDE) and the linear backward stochastic differential equation (BSDE). In our approach, the completion of squares method, together with the SRDE and the BSDE, allows us to construct an equivalent cost functional that is quadratic in the control $u$ , provided that an additional positive definiteness condition in terms of the SRDE holds. Then, the optimal control and the associated cost can be obtained by eliminating the quadratic term of $u$ in the equivalent cost functional. The additional positive definiteness condition is induced due to the indefiniteness of the cost parameters and the dependence of the control on the diffusion term. We verify the optimal solution by solving the stochastic HJB equation.

A kind of non-zero sum mixed differential game of backward stochastic differential equation

This paper is concerned with a non-zero sum mixed differential game problem described by a backward stochastic differential equation. Here the term “mixed” means that this game problem contains a deterministic control v_{1} of Player 1 and a random control process v_{2} of Player 2. By virtue of the classical variational method, a necessary condition and an Arrow’s sufficient condition for the mixed stochastic differential game problem are presented. A linear–quadratic mixed differential game problem is discussed, and the corresponding Nash equilibrium point is explicitly expressed by the solution of mean-field forward–backward stochastic differential equation. The most distinguishing feature, compared with the existing literature, is that the optimal state process of the linear–quadratic game satisfies a linear mean-field backward stochastic differential equation. Finally, a home mortgage and wealth management problem is given to illustrate our theoretical results.

Linear backward stochastic differential equations with Gaussian Volterra processes

Explicit solutions for a class of linear backward stochastic differential equations (BSDE) driven by Gaussian Volterra processes are given. These processes include the multifractional Brownian motion and the multifractional Ornstein-Uhlenbeck process. By an Ito formula, proven in the context of Malliavin calculus, the BSDE is associated to a linear second order partial differential equation with terminal condition whose solution is given by a Feynman-Kac type formula.

Integrability of exponential process and its application to backward stochastic differential equations

Abstract We consider the integrability problem of an exponential process with unbounded coefficients. The integrability is established under weaker conditions of Kazamaki type, which complements the results of Yong obtained under a Novikov type condition. As applications, we consider the solvability of linear backward stochastic differential equations (BSDEs) and market completeness, the solvability of a Riccati BSDE and optimal investment, all in the setting of unbounded coefficients.

Linear quadratic mean-field-game of backward stochastic differential systems

This paper is concerned with a dynamic game of N weakly-coupled linear backward stochastic differential equation (BSDE) systems involving mean-field interactions. The backward mean-field game (MFG) is introduced to establish the backward decentralized strategies. To this end, we introduce the notations of Hamiltonian-type consistency condition (HCC) and Riccati-type consistency condition (RCC) in BSDE setup. Then, the backward MFG strategies are derived based on HCC and RCC respectively. Under mild conditions, these two MFG solutions are shown to be equivalent. Next, the approximate Nash equilibrium of derived MFG strategies are also proved. In addition, the scalar-valued case of backward MFG is solved explicitly. As an illustration, one example from quadratic hedging with relative performance is further studied.

Backward-forward linear-quadratic mean-field games with major and minor agents

This paper studies the backward-forward linear-quadratic-Gaussian (LQG) games with major and minor agents (players). The state of major agent follows a linear backward stochastic differential equation (BSDE) and the states of minor agents are governed by linear forward stochastic differential equations (SDEs). The major agent is dominating as its state enters those of minor agents. On the other hand, all minor agents are individually negligible but their state-average affects the cost functional of major agent. The mean-field game in such backward-major and forward-minor setup is formulated to analyze the decentralized strategies. We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation (BFSDE) system. Next, we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method. Consequently, we obtain the decentralized strategies for major and minor agents which are proved to satisfy the e-Nash equilibrium property.

Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability

An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. In this paper, the existence of an optimal closed-loop strategy for the system (also called the closed-loop solvability of the problem) is characterized by the existence of a regular solution to the coupled two (generalized) Riccati equations, together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.

A general non-existence result for linear BSDEs driven by Gaussian processes

In this paper, we study linear backward stochastic differential equations driven by a class of centered Gaussian non-martingales, including fractional Brownian motion with Hurst parameter H∈(0,1)∖{12}. We show that, for every choice of deterministic coefficient functions, there is a square integrable terminal condition such that the equation has no solution.

Linear quadratic stochastic two-person zero-sum differential games in an infinite horizon

This paper is concerned with a linear quadratic stochastic two-person zero-sum differential game with constant coefficients in an infinite time horizon. Open-loop and closed-loop saddle points are introduced. The existence of closed-loop saddle points is characterized by the solvability of an algebraic Riccati equation with a certain stabilizing condition. A crucial result makes our approach work is the unique solvability of a class of linear backward stochastic differential equations in an infinite horizon.

Solutions to BSDEs Driven by Multidimensional Fractional Brownian Motions

We study more general backward stochastic differential equations driven by multidimensional fractional Brownian motions. Introducing the concept of the multidimensional fractional (or quasi-) conditional expectation, we study some of its properties. Using the quasi-conditional expectation and multidimensional fractional Itô formula, we obtain the existence and uniqueness of the solutions to BSDEs driven by multidimensional fractional Brownian motions, where a fixed point principle is employed. Finally, solutions to linear fractional backward stochastic differential equations are investigated.

H 2/H ∞ control problems of backward stochastic systems

This paper is concerned with the mixed H 2/H ∞ control problem for a new class of stochastic systems with exogenous disturbance signal. The most distinguishing feature, compared with the existing literatures, is that the systems are described by linear backward stochastic differential equations (BSDEs). The solution to this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique. Two equivalent expressions for the H 2/H ∞ control are presented. Contrary to forward deterministic and stochastic cases, the solution to the backward stochastic H 2/H ∞ control is no longer feedback of the current state; rather, it is feedback of the entire history of the state.

BSDEs driven by time-changed Lévy noises and optimal control

We study backward stochastic differential equations (BSDEs) for time-changed Lévy noises when the time-change is independent of the Lévy process. We prove existence and uniqueness of the solution and we obtain an explicit formula for linear BSDEs and a comparison principle. BSDEs naturally appear in control problems. Here we prove a sufficient maximum principle for a general optimal control problem of a system driven by a time-changed Lévy noise. As an illustration we solve the mean–variance portfolio selection problem.

Properties of solution of fractional backward stochastic differential equation

The backward stochastic differential equations (BSDEs) driven by fractional Brownian motion are studied. As an important tool, the quasi-conditional expectation is used. The general forms of Jensen’s inequality of quasi-conditional expectation are proved. For the linear BSDEs, their solutions are represented by using the quasi-conditional expectation. Moreover, the comparison theorem and comonotonic theorem of the solutions of linear BSDEs are derived.