Smoothness of Directed Chain Stochastic Differential Equations

TL;DR

This paper investigates the smoothness properties of solutions to directed chain stochastic differential equations, addressing the challenges posed by their chain structure using partial Malliavin derivatives.

Contribution

It introduces a novel approach employing partial Malliavin derivatives to analyze the smoothness of solutions in directed chain SDEs, overcoming limitations of classic methods.

Findings

Partial Malliavin derivatives enable smoothness analysis.

Classic Malliavin methods are insufficient for chain structures.

The approach extends understanding of stochastic chain systems.

Abstract

We study the smoothness of the solution of the directed chain stochastic differential equations, where each process is affected by its neighborhood process in an infinite directed chain graph, introduced by Detering et al. (2020). Because of the auxiliary process in the chain-like structure, classic methods of Malliavin derivatives are not directly applicable. Namely, we cannot make a connection between the Malliavin derivative and the first order derivative of the state process. It turns out that the partial Malliavin derivatives can be used here to fix this problem.