Study on Integration by Parts Formula Involving Malliavin Derivatives and Solutions to Delay SDE’S
Under suitably generic conditions, we derive an integration by parts formula using the space variable and the delay variable in this section. This formula is an expansion of the one found in Norris  as Theorem 2.3, but without the delay variable. The iterations of the integration by parts formula required to prove the smooth density conclusion require this generality. In this research, we derived a method for higher order Malliavin derivatives of delay stochastic differential equation solutions using parts integration. In some situations requiring densities of distributions of solutions of delay (as well as ordinary) stochastic differential equations with perhaps discontinuous preliminary information in a previous work, this integration by parts formula will be applied. This integration by parts formula can also be used to adapt Bally and Talay’s methods to account for both delay and ordinary SDEs.
Tagelsir A. Ahmed,
Department of Pure Mathematics, Faculty of Mathematical Science, University of Khartoum, P.O.Box 321, Khartoum, Sudan.
A. Van Casteren, Jan,
Department of Mathematics and Computer Science, University of Antwerp (UA), Middelheimlaan 1, 2020 Antwerp, Belgium.
Please see the link here: https://stm.bookpi.org/RAMRCS-V10/article/view/6269