Forward integration of bounded variation coefficients with respect to Hölder continuous processes

Abstract

In this article, we study the forward integral, in the Russo and Vallois sense, with respect to Hölder continuous stochastic processes Y with exponent bigger than 1∕2. Here, the integrands have the form f(Y), where f is a bounded variation function. As a consequence of our results, we show that this integral agrees with the generalized Stieltjes integral given by Zähle and that, in the case that Y is fractional Brownian motion, this forward integral is equal to the divergence operator plus a trace term, which is related to the local time of Y. Moreover, the definition of the forward integral allows us to obtain a representation of the solutions to forward stochastic differential equations with a possibly discontinuous coefficient and, as a consequence of our analysis, to figure out some explicit solutions.