Study on Integration by Parts Formula Involving Malliavin Derivatives and Solutions to Delay SDE’S

Abstract

We obtain in this section an integration by parts formula involving the space variable and the delay variable, under conditions sufficiently general. This formula is an extension of the formula which appears in Norris [1] as Theorem 2.3 without the delay variable. This generality is needed for the iterations of the integration by parts formula involved in proving the smooth density result. We developed an integration by parts formula for higher order Malliavin derivatives of delay stochastic differential equation solutions in this paper. This integration by parts formula will be used in certain applications involving densities of distributions of solutions of delay (as well as ordinary) stochastic differential equations with possibly discontinuous preliminary information in a squeal work. This integration by parts formula can also be used to modify the formulas in Bally and Talay's work to cover both delay and ordinary SDEs.