Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 29K / d / Solution

Let ${\mathcal{F}}_s=\sigma (W_r:0\le r\le s)$ and fix $c\in \mathbb{R}$. The martingale property between $s$ and $t$ givesFor fixed $s$, write the left-hand conditional expectation at time $t$ as $G(t)$. This integral equation has the unique solutionMultiplication by the ${\mathcal{F}}_s$-measurable factor $e^{-c{W}_s}$ therefore yields the conditional moment-generating functionThis is the moment-generating function of $N(0,t-s)$ and is deterministic. Hence $W_t-W_s$ has that normal law conditionally on ${\mathcal{F}}_s$, and its conditional law does not depend on ${\mathcal{F}}_s$; the increment is therefore independent of the past.

The process starts at zero by assumption and has continuous paths. The independent centred Gaussian increments just obtained complete the definition of Brownian motion. This proves the exponential test-function characterization of Brownian motion.

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