Joint distributions for stochastic functional differential equations

TL;DR

This paper establishes the existence of smooth joint distribution densities for solutions of stochastic functional differential equations with delay, using Malliavin calculus, and applies this to compute financial derivatives in delayed asset models.

Contribution

It provides new results on the smoothness of joint distributions for solutions of stochastic functional differential equations with delay, under ellipticity conditions.

Findings

Existence of smooth densities for solutions under elliptic conditions

Application to computing Greeks in delayed asset models

Use of Malliavin calculus for distribution analysis

Abstract

Consider stochastic functional differential equations, whose coefficients depend on past histories. The solution determines a non-Markov process. In the present paper, we shall obtain the existence of smooth densities for joint distributions of solutions, under the uniformly elliptic condition on the diffusion coefficients, via the Malliavin calculus. As an application, we shall study the computations of the Greeks on options associated with the asset price dynamics models with delayed effects.