abstract:
We consider the problem of approximating the expectation Ef(X(1)) of a function f of the solution X of a d-dimensional system of stochastic differential equations (SDE) at time point 1 based on finitely many evaluations of the coefficients of the SDE, the integrand f and their derivatives. We present a deterministic algorithm, which produces a quadrature rule by iteratively applying simplified weak Ito-Taylor steps together with strategies to reduce the diameter and the size of the support of a discrete measure.
We essentially assume that the coefficients of the SDE are s-times continuously differentiable and that the integrand f is r-times continuously differentiable. In the case r\leq (s-2)d/(d+2) we almost achieve an error of order min(r,s)/d in terms of the computational cost, which is optimal in a worst case sense. |