Computing conditional Wiener integrals of functionals of a general form

A numerical method of second order of accuracy for computing conditional Wiener integrals of smooth functionals of a general form is proposed. The method is based on simulation of Brownian bridge via the corresponding stochastic di¤erential equations (SDEs) and on ideas of the weak-sense numerical integration of SDEs. A convergence theorem is proved. Special attention is paid to integral-type functionals. A generalization to the case of pinned di¤usions is considered. Results of some numerical experiments are presented. Keywords. Conditional Wiener measure, Feynman path integrals, Brownian bridge, stochastic di¤erential equations, weak approximation, Monte Carlo technique, conditioned di¤usions. AMS 2000 subject classi…cation. 65C30, 28C20, 65C05, 60H35. 1 Introduction Let C 0;a;T;b be the set of all d-dimensional continuous vector-functions x(t) over [0; T ] satisfying the conditions x(0) = a; x(T ) = b: Consider the conditional Wiener integral J = Z Cd 0;a;T;b F (x( )) d T;b 0;a(x); (1.1) where F is a functional on C 0;a;T;b and T;b 0;a(x) is the conditional Wiener measure, corresponding to the Brownian paths X 0;a (t) with …xed initial and …nal points, i.e., it corresponds to the d-dimensional Brownian bridge from a at the time t = 0 to b at the time t = T: The integral (1.1) is to be understood in the sense of Lebesgue integral with respect to the measure T;b 0;a(x) and is taken over the set C d 0;a;T;b (see, e.g. [8, 24]). The importance of path integrals (1.1) for computing various quantities in quantum statistical mechanics is well known [5, 7, 8, 14, 23, 24]. For instance, the Feynman path integral of the form J =< aje TH jb > = Z exp Z T 0 m _ x(t) 2 V (x(t)) dt Dx(t); H = 1 2 + V;