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Let ${\psi}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth andcontinuous real function and $\psi\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let${B}(t)$ be a standard Brownian motion defined with respect to a probabilityspace $(\Omega,\mathscr{F},{\mathsf{P}})$ and where $d{B}(t)={\xi}(t)dt$ and$t\in\mathbb{R}^{+}$. The process $\xi(t)$ is a Gaussian white noise withexpectation $\mathbf{\mathsf{E}}~\xi(t)=0$ and with covariance${\mathsf{E}}~\xi(t)\xi(s)=\delta(t-s)$. The Dolean-Dades stochasticexponential ${Z}(t)$ is the solution to the linear stochastic differentialequation 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 expansionmethod, and the moment and cumulant generating functions for $\xi(t)$, it isshown 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 $\phi(t)=\int_{0}^{t}|\psi(u)|^{2}du$ ismonotone increasing and is a submartingale for all $p>1$.

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