Forward - backward stochastic differential equations with random coeffcients and applications to finance

Abstract

The first part of this thesis studies forward and backward versions of the random Burgers equation (RBE) with stochastic coeffcients. First, the celebrated Cole-Hopf transformation reduces the forward RBE to a forward random heat equation (RHE) that can be treated pathwise. Next we provide a connection between the backward Burgers equation and a system of forward backward stochastic differential equations (FBSDEs). Exploiting this connection, we derive a generalization of the Cole-Hopf transformation which links the backward RBE with the backward RHE and investigate the range of its applicability. Stochastic Feynman- Kac representations for the solutions are provided. Explicit solutions are constructed and applications to stochastic control and mathematical finance are discussed.