A Class of Approximations of Brownian Motion

Let B(t) = (B\t)i B(t), • • , B(t)} be a ^-dimensional Brownian motion and let {Bn(i) = (Bn (t),Bn\f),-~,Bn*(i))} be a sequence of approximations to B(t). We assume that the sample paths of Bn(t) are continuous and piecewise smooth for each n and Bn(t) converges to B(t). Let u(x) be a twice continuously differentiate function on R whose partial derivatives of order <^2 are all bounded. In the one-dimensional case E.Wong and M. Zakai [5] showed that I u(Bn(s))dBn(s) conJo verges to I u (B (s)) odB(s) where the symbol o denotes the symmetric Jo stochastic integral of Stratonovich (K. Ito [2]). They also dealt with the convergence of the more general functional of Bn(-)9 ([6]). In the two-dimensional case P. Levy [3] showed that S(t;n)= \ (B^(s)Jo dBJ(s) — B»(s)dBJ(s))/2 converges to the stochastic integral S(t) = (\B(s)°dff(s) -B(s}°dB(s')}/2 if (Bn(t)} is a sequence of polygonal Jo approximations to B ( f ) . E. J. McShane [4], on the other hand, gave an example of the sequence {Bn(t}} of approximations to B(t) such that S(t;n} converges to S(f)-\-t/n. In this paper we treat systematically a class of approximations of Brownian motion including McShane's example. In Section 2 we state the main results of the paper. We consider a sequence of Stieltjes integrals of the form In(ii) — \ u(Bn(s))dBn (5). First we will give some Jo conditions under which In(u) converges in the quadratic-mean sense. It