Backward Stochastic Differential Equations and Stochastic Controls: A New Perspective

It is well known that backward stochastic differential equations (BSDEs) stem from the study on the Pontryagin type maximum principle for optional stochastic control. A solution of a BSDE hits a given terminal value (which is a random variable) by virtue of an additional martingale term and an indefinite 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 specifically, associated with a BSDE a new stochastic control problem is introduced with the same dynamics but a definite initial state. The martingale term in the origional BSDE is regarded as the control and the objective is to minimize the second moment of the difference 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 Back-Scholes model, where an optimal feedback control is obtained expicitly 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 take into account.