By formulating and extending some formulas and results on Malliavin calculus and ordinary stochastic differential equations to include delay stochastic differential equations as well as ordinary SDE's (see [1–11]), we have taken a step forward towards integration by part of higher order Malliavin derivatives. We've also defined what the Malliavin derivatives and densities of distributions of the solutions process for delay stochastic differential equations signify in this section. In general, we may claim that our work expands Norris' book's first three chapters to cover both delay and ordinary SDEs; see Theorems 2.3, 3.1, and 3.2 in [12]. We will also demonstrate in a follow-up publication to this effort that the solution method has a smooth density distribution. In addition, using Malliavin derivatives of higher order, we shall develop an integration by parts formula. Note that the delay SDE ([E: V 3]) is a Norris extension of the SDE (3.3) that includes both delay and conventional SDEs. This may be shown by examining just the entries in ([E: V 3]) that include derivatives of the coefficients with respect to the space variable while including none with respect to the delay variable. We are automatically in the Norris case of SDEs if we do this. The SDEs (2.31), (2.32), and (2.33) in Norris [12] are equal to the SDEs (3.1), (3.2), and (3.3) in Norris [12]. As a result, we can see that our delay stochastic differential equations (2.28), (2.15), and (2.30) expand Norris' SDEs (3.1), (3.2), and (3.3), and contain both delay and ordinary SDEs.
Author(S) Details
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.
View Book:- https://stm.bookpi.org/NRAMCS-V2/article/view/6787
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