Second-order schemes for solving decoupled forward backward stochastic differential equations (Q2254815)

Three numerical schemes are derived for solving decoupled Itô forward backward stochastic differential equations (FBSDE) of the form \[ \begin{aligned} X_t &= X_0+ \int^t_0 b(s,X_s)\,ds+ \int^t_0 \sigma(s, X_s)\,dW_s,\\ Y_t &= \varphi(X_\Upsilon)+ \int^T_t f(s,X_s, Y_s, Z_s)\,ds- \int^T_t Z_s dW_s,\end{aligned} \] where \(W_t\) is a \(d\)-dimensional Brownian motion. For two of the schemes theorems giving error estimates are proved that establish a convergence rate of 2.0 for the approximations of \(Y_t\) and 1.5 for the approximations of \(Z_t\). Results of numerical experiments for two FBSDE are summarized and indicate that convergence rates for all three schemes are of order 1 if the Euler method or Milstein method, which have weak order 1, are used to solve the forward SDE and are of order 2 if an order 2 weak Itô-Taylor method is used to solve the forward SDE. In these experiments the computational costs are much less when the Euler method is used, but the best efficiency is obtained by using the Itô-Taylor method.