Applications of the Integration by Parts Formula Involving Malliavin Derivative of the Solution Process and it’s Inverse
In the current work, we have developed a few applications of the integration by parts formula that allow us to set the groundwork for the investigation of the regularity characteristics of the distributions of the stochastic delay equation solution process. In this work, we review our fundamental Delay SDE (1.1), and then we define what we mean by the space flow of the solution process, which is the Malliavin derivative of the Delay SDE (1.1) solution process. Finally, we formulate the corresponding Delay SDE of the space flow, as shown in equation (2.12) and its inverse, as shown in equation (2.13) (2.13).
In equations (2.6), (2.7), and (2.8), we have introduced the stochastic differentials for a clear formulation of the SDE of the space flow and its inverse (2.8). See also the Delay SDE’s in this work, which deal with the space flow and its inverse, respectively, at 2.16 and 2.17.
Author (s) Details:
Tagelsir A. Ahmed,
Department of Pure Mathematics, Faculty of Mathematical Science, University of Khartoum, B. O. Box 321, Khartoum, Sudan.
Jan A. Van Casteren,
Department of Mathematics and Computer Science, University of Antwerp (UA), Middelheimlaan 1, 2020 Antwerp, Belgium.
Please see the link here: https://stm.bookpi.org/NRAMCS-V5/article/view/7490
Keywords: Stochastic differential equations, malliavin calculus, euler scheme for delay SDE’s, integration by parts, mDensities of distributions.