Two-parameter stochastic calculus and Malliavin's integration-by-parts formula on Wiener space

TL;DR

This paper develops a two-parameter stochastic calculus framework to prove Malliavin's integration-by-parts formula on Wiener space and demonstrates that solutions to certain stochastic differential equations are two-parameter semimartingales.

Contribution

It introduces a novel two-parameter stochastic calculus approach to establish Malliavin's integration-by-parts formula and characterizes solutions of specific stochastic differential equations as two-parameter semimartingales.

Findings

Proved Malliavin's integration-by-parts formula using two-parameter stochastic calculus.

Showed solutions to certain SDEs driven by two-parameter semimartingales are themselves two-parameter semimartingales.

Established a connection between two-parameter stochastic calculus and Wiener space analysis.

Abstract

The integration-by-parts formula discovered by Malliavin for the Ito map on Wiener space is proved using the two-parameter stochastic calculus. It is also shown that the solution of a one-parameter stochastic differential equation driven by a two-parameter semimartingale is itself a two-parameter semimartingale.