Backward Stochastic Di erential Equations and Stochastic Controls: A New Perspective

It is well known that backward stochastic di erential 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 additional martingale term and an inde nite initial state. This paper attempts to view the relation between BSDEs and stochastic controls from s new perspective by interpreting BSDEs as some stochastic optimal control problems. More speci cally, associated with a BSDE a new stochastic control problem is introduced with the same dynamics but a de nite 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 di erence between the terminal state and the given terminal value. This problem is solved in a closed form by the stochastic linear-quadratic theory developed recently. The general result is then applied to the Black-Scholes model, where an optimal feedback control is obtained explicitly in terms of the option price. Finally, a modi ed model is investigated where the di erence between the state and the expectation of the given terminal value at any time is take into account. Abbreviated Title. BSDEs and Stochastic Controls AMS 1991 subject classi cations.