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

Abstract

Let $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ 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 $(\Omega ,\mathcal{F},\mathsf{P})$ and where $dB(t)=\xi (t)dt$ and $t\in {\mathbb{R}}^{+}$. The process $\xi (t)$ is a Gaussian white noise with expectation $\mathsf{E}\xi (t)=0$ 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 $dZ(t)=\psi (t)Z(t)dB(t)=\psi (t)Z(t)\xi (t)dt$. 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}ψ(u)d{B}(u) -\frac{1}{2}\int_{0}^{t}|ψ(u)|^{2}du\right)=1\nonumber \end{align} provided that $\exp ({\int }_0^t|\psi (u){|}^{2}du)<\infty$ for all $t>0$. However, $\mathsf{E}[|Z(t){|}^p]=\exp (\frac{1}{2}p(p-1)\varphi (t))$, if $\varphi (t)={\int }_0^t|\psi (u){|}^{2}du$ is monotone increasing and is a submartingale for all $p>1$.

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