A Stratonovich integral for anticipating processes

Abstract

A stochastic integral for anticipating integrands was introduced by Ayed and Kuo in 2008. Riemann–Stieltjes sums were considered, where the adapted part of the integrand was evaluated at the left endpoints of the subintervals, while the instantly independent part was evaluated at the right endpoints. Since then, many results have been proved, such as formulas for differentials. In this paper, the Stratonovich counterpart of the Ayed–Kuo integral is investigated. In its simplest version, it is proved that, analogously to the classical stochastic integration theory for adapted processes, the fundamental theorem of calculus holds. Consequences, extensions, and limitations are discussed in detail.