A Representation Theorem for Smooth Brownian Martingales

TL;DR

This paper establishes a representation theorem for smooth Brownian martingales, expressing them as exponentials involving their Malliavin derivatives, extending semi-group theory to path-dependent PDEs, with implications for numerical methods.

Contribution

It introduces a novel exponential representation for Brownian martingales under smoothness conditions, linking it to Malliavin calculus and path-dependent PDEs, and provides a series expansion for practical computation.

Findings

Representation of martingales as exponential of future values

Explicit series expansion similar to Dyson series

Numerical approximation via backward Taylor expansion

Abstract

We show that, under certain smoothness conditions, a Brownian martingale at a fixed time can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times the Malliavin derivative. This result can also be seen as a generalization of the semi-group theory of parabolic partial differential equations to the parabolic path-dependent partial differential equations introduced by Dupire (2009) and Cont and Founi\'e (2011). The exponential operator can be calculated explicitly in a series expansion, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result is proved by a passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The latter expansion can also be used for numerical calculations.