Backward Stochastic Differential Equations

In this chapter, we introduce backward stochastic differential equations (in short BSDEs). Recently, first order BSDEs (1-BSDEs) have attracted attention in the mathematical finance community, see El Karoui et al. [101] for references and a review of applications. One of the key features of 1-BSDEs is that they provide a probabilistic representation of solutions of nonlinear parabolic PDEs, generalizing the Feynman-Kac formula. However, the corresponding PDEs cannot be nonlinear in the second order derivative and are therefore connected to HJB equations with no control on the diffusion term. In [81], Cheridito et al. provide a stochastic representation for solutions of fully nonlinear parabolic PDEs by introducing a new class of BSDEs, the so-called second order BSDEs (in short 2-BSDEs). We first present a short accessible introduction that gives the main definitions and results, explains why 1-BSDEs provide probabilistic representations of nonlinear PDEs, and addresses the question of numerical simulation. Then we deal with reflected 1-BSDEs and their application to the valuation of American-style options. Eventually we introduce 2-BSDEs.