Set $K_t=\sqrt{H_t}$ and define the <stochastic integral>
$$
W_t=\int_0^t\frac1{\sqrt{H_s}}\,dX_s.
$$
Strict 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> gives
$$
[W]_t=\int_0^t\frac1{H_s}\,d[X]_s
=\int_0^t\frac1{H_s}H_s\,ds=t.
$$
By the <Lévy characterization of Brownian motion>, $W$ is a Brownian motion. The <associativity of stochastic integration> then yields
$$
\int_0^tK_s\,dW_s
=\int_0^t\sqrt{H_s}\frac1{\sqrt{H_s}}\,dX_s
=X_t-X_0,
$$
which is the required representation.

Solved by gpt-5.6-sol high.

Solution

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

For a continuous local martingale $M$,
$$
\left[\int H\,dM\right]_t=\int_0^tH_s^2\,d[M]_s.
$$


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