Weak convergence to the multiple Stratonovich integral

[摘要] We have considered the problem of the weak convergence, as epsilon tends to zero, of the multiple integral processes {integral (t)(0) ... integral (t)(0) f(t(1),...,t(n)) d eta (epsilon)(t(1)) ... d eta (epsilon)(t(n)), t epsilon [0,T]} in the space C-0([0,T]), where f epsilon L-2([0,T](n)) is a given function, and {eta (epsilon)(t)}(epsilon >0) is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n greater than or equal to2 and f(t(1),..., t(n)) = 1({t1 < t2 <...< tn}), we cannot expect that these multiple integrals converge to the multiple Ito-Wiener integral of f, because the quadratic variations of the (epsilon) are null. We have obtained the existence of the limit for any {eta (epsilon)}, when f is given by a multimeasure, and under some conditions on {eta (epsilon)} when f is a continuous function and when f(t(1),..., t(n)) = f(1)(t(1)) f(n)(t(n))1({t1 < t2 <...< tn}), with f(i) L-2([0, T]) for any i = 1,...,n. In all these cases the limit process is the multiple Stratonovich integral of the function f. (C) 2000 Elsevier Science B.V. All rights reserved.

[发布日期] 2000-12-01

[关键词] weak convergence;multiple Stratonovich integral;multimeasure;donsker approximations