Abstract

Forward-Backward Stochastic Differential equations (FBSDEs) have been successfully applied to problems in mathematical finance, such as hedging of contingent claims and modelling stock sale-advertising responses. An example of a simple problem is

dX(t) = {(X(t)) / ((Z(t)-Y(t))² + 1 )} dt + X(t)dW(t)
dY(t) = {(Z(t)) / ((Z(t)-Y(t))² + 1 )} dt + Z(t)dW(t)
X(0) = x
Y(T) = X(T)

whose an analytical solution is X = Y = Z = x.exp{(t/2+W(t))}, t in [0,T].

The problem is difficult to solve since we look for an adapted solution (X,Y,Z) where Z is still implicit in the equation. Some methods can solve this problem, including a four-step scheme where the adapted solution is obtained by solving a quasilinear partial differential equation. Instead of following that approach, we use waveform relaxation which requires us to solve problems of the form

dY^k+1(t) = h^k(t) dt + Z^k+1(t) dW(t)

at each iteration. Numerical results show improved convergence when the technique of windowing is applied along the region of integration.