Applications of the Integration by Parts Formula Involving Malliavin Derivative of the Solution Process and it's Inverse | Chapter 13 | Novel Research Aspects in Mathematical and Computer Science Vol. 5

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

Assessment of Densities of Distributions of Solutions to Delay Stochastic Differential Equations with Discontinuous Initial Data (Part I) | Chapter 01 | Novel Research Aspects in Mathematical and Computer Science Vol. 3

 The goal of this research is to show that the Integration by Parts formula we developed in this paper can be used to expand all of Bally and Talay's formulae (in [1]) to include delay SDEs as well as SDEs. This indicates that this research may be used to determine the rate of convergence of the density of the distribution of the solution process for both delay and conventional SDEs. We developed an integration by parts formula employing Malliavin derevatives of delay (functional) SDE solutions (see equation) (1.1). The integration by parts formula we developed is actually an expansion of the integration by parts formula that includes both delay and conventional SDEs. The incorporation of We have demonstrated that the components formula may be utilised to expand Bally and Talay's formulations to include both delay and regular 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-V3/article/view/6808

Determining the Densities of Distributions of Solutions to Delay Stochastic Differential Equations with Discontinuous Initial Data | Chapter 01 | Novel Research Aspects in Mathematical and Computer Science Vol. 2

 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

Study on Integration by Parts Formula Involving Malliavin Derivatives and Solutions to Delay SDE’S| Chapter 1 | Recent Advances in Mathematical Research and Computer Science Vol. 10

  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 [1] 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.

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.

Please see the link here: https://stm.bookpi.org/RAMRCS-V10/article/view/6269