Lithuanian Mathematical Journal, Vol. 33, No. 4, 1993 GAUSSIAN APPROXIMATIONS OF BROWNIAN MOTION IN A STOCHASTIC INTEGRAL V. MackeviSius and B. Zibaitis Abstract--Given a Brownian motion B, Gaussian approximations B 6, ~ > 0, of the form Bt ~ = .f: fR g6(u' s)dB~ du, t > O, including polygonal and mollifier approximations, are considered. A r limit theorem is proved for the integrals fo Xt dB~t as ~ ~ O. In particular, in the case of symmetric T kernels K 6 the limit is the Fisk-Stratonovich integral fo Xt o dBt. 1. INTRODUCTION AND FORMULATION OF RESULT Symmetric (Fisk-Stratonovich) stochastic integrals f0 T Xt o dBt often appear as the limits of ordinary integrals f: X6t dB~t when the Brownian motion B is approximated by finite variation processes B ~, ~ > 0, and the integrand process X is replaced by appropriately chosen approximations X ~, 6 > 0 (cf., e.g., Wong- Zakai [8, 9], nalakrishnan [1], Ikeda-Watanabe [3], Mackevi~ius [6]). A problem of investigating the integrals f[ Xt dBSt, similar at first sight has significant peculiarities. Take, for example, the arbitrary continuous semimartingale X and the arbitrary piecewise continuous non-anticipating approximation B ~, 5 > 0, such that supt<T IB~t - Bt] --+ 0 in the mean-square sense. Then T T T T 0 0 0 0 The latter is equal neither to the It5 integral f[ Xt dBt, nor to the symmetric integral T T 1 (x, /XtodBt 0 0 However, on the other hand, we have proved in [7, 10] that T T o o as 6--+0 in the two following "anticipating" cases: a) B 8 are the polygonal (piecewise linear) approximations; b) B * are mollifier (convolution type) approximations with a symmetric (even) smoothing function ~o. If, on the contrary, Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Vilnius Pedagogical University, Studentu 39, 2034 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 33, No. 4, pp. 508-526, October-December, 1993. Original article submitted March 4, 1993. 0363-1672/93/3304-0393512.50 C) 1994Plenum Publishing Corporation 393