Let $ψ:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth and continuous real function and $ψ\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let ${B}(t)$ be a standard Brownian motion defined with respect to a probability space $(Ω,\mathscr{F},{\mathsf{P}})$ and where $d{B}(t)=ξ(t)dt$ and $t\in\mathbb{R}^{+}$. The process $ξ(t)$ is a Gaussian white noise with expectation $\mathbf{\mathsf{E}}~ξ(t)=0$ and with covariance ${\mathsf{E}}~ξ(t)ξ(s)=δ(t-s)$. The Dolean-Dades stochastic exponential ${Z}(t)$ is the solution to the linear stochastic differential equation describing a geometric Brownian motion such that $d{Z}(t)=ψ(t){Z}(t)d{B}(t)=ψ(t){Z}(t)ξ(t)dt$. Using a cluster expansion method, and the moment and cumulant generating functions for $ξ(t)$, 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 $\exp\big(\int_{0}^{t}|ψ(u)|^{2}du\big)<\infty$ for all $t>0$. However, ${\mathsf{E}}\big[|{Z}(t)|^{p}\big] =\exp(\tfrac{1}{2}p(p-1)φ(t))$, if $φ(t)=\int_{0}^{t}|ψ(u)|^{2}du$ is monotone increasing and is a submartingale for all $p>1$.