Assessment of Densities of Distributions of Solutions to Delay Stochastic Differential Equations with Discontinuous Initial Data (Part I)

The objective of this study is that The Integration by parts formula which we have established in this work is needed to extend all the formulas by Bally and Talay (in [1]) to include delay SDE's as well as SDE's. This means that this work is very useful in finding the rate of convergence of the density of the distribution of the solution process of delay SDE's as well as ordinary SDE's. We have established an integration by parts formula involving Malliavin derevatives of solutions to the delay (functional) SDE’s, See equation (1.1). The integration by parts formula which we have established is in fact an extension of the integration by parts formula to include delay SDE’s as well as ordinary SDE’s. The integration by parts formula which we have established can be used to extend the formulas in work by Bally and Talay to include delay SDE’s as well as ordinary SDE’s