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

Apply Itô formula to $Z_t^{(A)}=\exp (A{X}_t-A^{2}t/2)$. Its semimartingale decomposition isThe second term is a continuous finite-variation process. Since $Z^{(A)}$ is assumed to be a local martingale, uniqueness of the semimartingale decomposition makes this term identically zero. Both $A$ and $Z^{(A)}$ are nonzero, so the quadratic variation of $X$ is $[X{]}_t=t$.

The Lévy characterization of Brownian motion now says that $X_t-X_0$ is a Brownian motion. Consequentlyis a constant multiple of an exponential Brownian martingale. It is therefore a true martingale for every $a\in \mathbb{R}$.

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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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Use first the test function $f(x)=x$. The assumed martingale problem says thatis a continuous local martingale. Next use $f(x)=x^2$ to see thatis a local martingale. On the other hand, Itô formula applied to $X=X_0+\int b(X_s)ds+Y$ shows thatis a local martingale. Their difference is both a continuous local martingale and a finite-variation process, so

Part (b), with $H_s=\sigma (X_s{)}^2>0$, supplies a Brownian motion $W$ such thatThereforeso $X$ is a weak solution of a stochastic differential equation to the stated equation.

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