An approximate method via Taylor series for stochastic functional differential equations (Q1043908)

Using Taylor expansions of \(f\) and \(g\), approximate solutions are generated for a stochastic functional differential equation of the form \[ dx(t)= f(x_t, t)\,dt+ g(x_t,t)\,dw(t), \] where \(f\) and \(g\) are functionals and \(w\) is standard Brownian motion. Then it is proved that the approximate solutions converge pathwise to the actual solution. An error bound is derived that relates the maximum error to the degrees of the two Taylor expansions used.