Relationship Between Backward Stochastic Differential Equations and Stochastic Controls: A Linear-Quadratic Approach

It is well known that backward stochastic differential equations (BSDEs) stem from the study on the Pontryagin type maximum principle for optimal stochastic controls. A solution of a BSDE hits a given terminal value (which is a random variable) by virtue of an it additional martingale term and an indefinite initial state. This paper attempts to explore the relationship between BSDEs and stochastic controls by interpreting BSDEs as some stochastic optimal control problems. More specifically, associated with a BSDE, a new stochastic control problem is introduced with the same dynamics but a definite given initial state. The martingale term in the original BSDE is regarded as the control, and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the BSDE. This problem is solved in a closed form by the stochastic linear-quadratic (LQ) theory developed recently. The general result is then applied to the Black--Scholes model, where an optimal mean-variance hedging portfolio is obtained explicitly in terms of the option price. Finally, a modified model is investigated, where the difference between the state and the expectation of the given terminal value at any time is taken into account.

[1]  J. Bismut.  An Introductory Approach to Duality in Optimal Stochastic Control , 1978 .

[2]  Robert J. Elliott,et al.  Mathematics of Financial Markets , 1999 .

[3]  W. Wonham.  On a Matrix Riccati Equation of Stochastic Control , 1968 .

[4]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[5]  Jin Ma,et al.  Solvability of forward-backward SDEs and the nodal set of Hamilton-Jacobi-Bellman Equations , 1993 .

[6]  Larry G. Epstein,et al.  Stochastic differential utility , 1992 .

[7]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[8]  Majin,et al.  SOLVABILITY OF FORWARD-BACKWARD SDES AND THE NODAL SET OF HAMILTON-JACOBI-BELLMAN EQUATIONS , 1995 .

[9]  Jiongmin Yong,et al.  Approximate Solvability of Forward—Backward Stochastic Differential Equations , 2002 .

[10]  H. Kunita,et al.  On Square Integrable Martingales , 1967, Nagoya Mathematical Journal.

[11]  Jiongmin Yong,et al.  Linear Forward—Backward Stochastic Differential Equations , 1999 .

[12]  R. E. Kalman,et al.  Contributions to the Theory of Optimal Control , 1960 .

[13]  R. C. Merton,et al.  Optimum consumption and portfolio rules in a continuous - time model Journal of Economic Theory 3 , 1971 .

[14]  Xun Yu Zhou,et al.  Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs. II , 2000, SIAM J. Control. Optim..

[15]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[16]  J. Ma,et al.  Forward-Backward Stochastic Differential Equations and their Applications , 2007 .

[17]  J. Yong,et al.  Solving forward-backward stochastic differential equations explicitly — a four step scheme , 1994 .

[18]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[19]  H. Pham,et al.  Mean‐Variance Hedging and Numéraire , 1998 .

[20]  Huyên Pham,et al.  Dynamic programming and mean-variance hedging , 1999, Finance Stochastics.

[21]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .