A Cluster Expansion Proof That The Stochastic Exponential Of A Brownian Motion Is A Martingale

TL;DR

This paper proves, using a cluster expansion method, that the stochastic exponential of a Brownian motion is a martingale, confirming classical criteria and providing new insights into its moment properties.

Contribution

It introduces a cluster expansion approach to establish the martingale property of the stochastic exponential of Brownian motion, reproducing and extending Novikov's criteria.

Findings

The stochastic exponential satisfies Novikov's criteria for martingales.

The moments of the exponential are explicitly characterized.

The method offers a new proof technique for martingale properties.

Abstract

Let ${\psi}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth and continuous real function and $\psi\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let ${B}(t)$ be a standard Brownian motion defined with respect to a probability space $(\Omega,\mathscr{F},{\mathsf{P}})$ and where $d{B}(t)={\xi}(t)dt$ and $t\in\mathbb{R}^{+}$. The process $\xi(t)$ is a Gaussian white noise with expectation $\mathbf{\mathsf{E}}~\xi(t)=0$ and with covariance ${\mathsf{E}}~\xi(t)\xi(s)=\delta(t-s)$. The Dolean-Dades stochastic exponential ${Z}(t)$ is the solution to the linear stochastic differential equation describing a geometric Brownian motion such that $d{Z}(t)=\psi(t){Z}(t)d{B}(t)=\psi(t){Z}(t)\xi(t)dt$. Using a cluster expansion method, and the moment and cumulant generating functions for $\xi(t)$, it is shown that ${Z}(t)$ is a martingale. The original Novikov criteria for ${Z}(t)$ being a true martingale are reproduced and exactly satisfied, namely that \begin{align} {\mathsf{E}}\mathrm{Z}(t)={\mathsf{E}}\exp\left(\int_{o}^{t}{\psi}(u)d{B}(u) -\frac{1}{2}\int_{0}^{t}|{\psi}(u)|^{2}du\right)=1\nonumber \end{align} provided that $\exp\big(\int_{0}^{t}|{\psi}(u)|^{2}du\big)<\infty$ for all $t>0$. However, ${\mathsf{E}}\big[|{Z}(t)|^{p}\big] =\exp(\tfrac{1}{2}p(p-1)\phi(t))$, if $\phi(t)=\int_{0}^{t}|\psi(u)|^{2}du$ is monotone increasing and is a submartingale for all $p>1$.