Backward Stochastic Differential Equations in Financial Mathematics

Abstract: A backward stochastic differential equation (BSDE) is an SDE of the form $-dY_t = f(t,Y_t,Z_t)dt - Z_t^*dW_t;\ Y_T = \xi$. The subject of BSDEs has seen extensive attention since their introduction in the linear case by Bismut (1973) and in the general case by Pardoux and Peng (1990). In contrast with deterministic differential equations, it is not enough to simply reverse the direction of time and treat the terminal condition as an initial condition, as we would then run into problems with adaptedness. Intuitively, our "knowledge" at time $t$ consists only of what has happened at all times $s \in [0,t]$, and we cannot reverse the direction of time whilst keeping this true. The layout of this essay is as follows: In Section 1 we introduce BSDEs and go over the basic results of BSDE theory, including two major theorems: the existence and uniqueness of solutions and the comparison theorem. We also introduce linear BSDEs and the notion of supersolutions of a BSDE. In Section 2 we set up the financial framework in which we will price European contingent claims, and prove a result about the fair price of such claims in a dynamically complete market. In Section 3 we extend the theory of linear BSDEs to include concave BSDEs, and apply this to pricing claims in more complicated market models. In Section 4 we take a look at utility maximisation problems, and see how utilising BSDE theory allows for a relatively simple and neat solution in certain cases.