Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 202 / 1 / b / Solution

Set $K_t=\sqrt{H_t}$ and define the stochastic integralStrict positivity and predictability of $H$ make the integrand locally admissible. The process $W$ is a continuous local martingale starting from zero, and the quadratic variation of a stochastic integral givesBy the Lévy characterization of Brownian motion, $W$ is a Brownian motion. The associativity of stochastic integration then yieldswhich is the required representation.

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