Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 28J (Stochastic Financial Models)

Relative to a filtration $({\mathcal{F}}_t)$, a martingale is an adapted integrable process satisfying $\mathbb{E}[M_t\mid {\mathcal{F}}_s]=M_s$ for $s\le t$. The martingale convergence theorem gives an almost-sure finite limit for a nonnegative martingale, with $\mathbb{E}[M_{\infty }]\le \mathbb{E}[M_0]$ by Fatou's lemma. In continuous time take the usual right-continuous version and filtration hypotheses. Convergence in $L^1$, and preservation of expectation at infinity, additionally require uniform integrability.

A standard Brownian motion starts at zero, has continuous paths, and has independent increments $B_t-B_s\sim N(0,t-s)$. By the Brownian reflection principle, for $a>0$,Hence every positive integer level is reached with probability one. Intersecting these countably many probability-one events proves $\boxed{\underset{t\ge 0}{\sup }{B}_t=\infty \text{almost surely}.}$

Independence and the normal exponential moment give $\mathbb{E}[S_t\mid {\mathcal{F}}_s]=S_s\exp [(\mu +{\sigma }^2/2)(t-s)]$. ThusFor $\sigma \ne 0$, $B_t/t\to 0$ almost surely, so $\log S_t/t\to -{\sigma }^2/2$ and $S_t\to 0$ almost surely. Nevertheless $\mathbb{E}{S}_t=e^{x_0}$ for every finite $t$, whereas $\mathbb{E}{S}_{\infty }=0$. This Exponential martingale for Brownian motion is not uniformly integrable, so it does not converge in $L^1$. If $\sigma =0$, the process is instead constantly $e^{x_0}$.