Determining the Densities of Distributions of Solutions to Delay Stochastic Differential Equations with Discontinuous Initial Data

Abstract

In the present work we have gone a step forward towards integration by part of higher order Malliavin derivatives 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]). Here we have also stated clearly what we mean by the Malliavin derivatives and densities of distributions of the solutions process for delay stochastic differential equations which we are considering. Generally speaking we can say that our work extends the first three chapters of the work by Norris to include delay SDE’s as well as ordinary SDE’s; see Theorems 2.3, 3.1 and 3.2 in [12]. We will also show in a Sequal paper to this work that the distribution of the solution process has smooth density. Also we will establish an integration by parts formula involving Malliavin derivatives of higher order. Observe that the delay SDE ([E: V 3]) is an extension of the SDE (3.3) in Norris to include delay SDE’s as well as ordinary SDE’s. We can see this by considering only the terms in ([E: V 3]) which include derivatives of the coefficients with respect to the space variable and in the same time it contain no derivative with respect to the delay variable. If we do this then we are automatically in the Norris case of SDE’s. Observe that the SDE’s (2.31), (2.32) and (2.33) are equivalent to the SDE’s (3.1), (3.2) and (3.3) in Norris [12] respectively. Thus we can see that our delay stochastic differential equations (2.28), (2.15) and (2.30) in fact extend the SDE’s (3.1), (3.2) and (3.3) in Norris; they include the case of delay as well as ordinary SDE’s.