Solving Backward Stochastic Differential Equations Using the Cubature Method: Application to Nonlinear Pricing

We are concerned with the numerical solution of a class of backward stochastic differential equations (BSDEs), where the terminal condition is a function of $X_T$, where $X=\{X_t,t\in [0,T]\}$ is the solution to a standard stochastic differential equation (SDE). A characteristic of these type of BSDEs is that their solutions $Y=\{Y_t,t\in [0,T]\}$ can be written as functions of time and $X$, $Y_t=\vartheta_t(X_t)$. Moreover, the function $\vartheta_t$ can be represented as the expected value of a functional of $X$. Therefore, since the forward component $X_{t}$ is „known” at time $t$, the problem of estimating $Y_{t}$ amounts to obtaining an approximation of the expected value of the corresponding functional. The approximation of the solution of a BSDE requires an approximation of the law of the solution of the SDE satisfied by the forward component. We introduce a new algorithm, combining the Euler style discretization for BSDEs and the cubature method of Lyons and Victoir [Proc. R. Soc. Lond. Ser. A Mat...