Two results on multiple Stratonovich integrals

Abstract

Formulae connecting the multiple Stratonovich integrals with single Ogawa and Stratonovich integrals are derived. Multiple Riemann-Stieltjes integrals with respect to certain smooth approximations of the Wiener process are considered and it is shown that these integrals converge to multiple Stratonovich integrals as the approximation converges to the Wiener process. In an important work Hu and Meyer (1987) introduced a new multi- ple stochastic integral with respect to a Wiener process, called the multiple Startonovich integral (MSI), which is in general different from the usually studied multiple Wiener-Ito integral. However, Hu and Meyer only offered some rather informal definitions and proofs. Johnson and Kallianpur (1993) gave a rigor- ous definition for the MSI (the term MSI was not used in their work), denoted in this work by δp(·), and gave necessary and sufficient conditions for its exis- tence. Recently, multiple Stratonovich integrals have been applied to problems in asymptotic statistics and nonlinear filtering (cf. Budhiraja and Kallianpur (1995, 1996)). In this work we study some properties of multiple Stratonovich integrals which also give some justification for the appearance of the name 'Stratonovich' in the integral. We recall that one of the important properties of multiple Wiener-Itoi nte- grals of symmetric kernels is that they can be expressed as iterated (indefinite) Ito-integrals, by which we mean that if fp ∈ L 2(0, 1) p (the class of real val- ued, square integrable, symmetric functions defined on (0, 1) p ) then the multiple