Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 1 / b

The existence assertion is false under the hypotheses printed in the source. No measurability of $\theta$ is specified. Even requiring $\theta$ to be a Borel measurable function does not suffice: a coefficient can be positive at every point while approaching zero too rapidly near the starting point. This is the obstruction described by pointwise positivity does not ensure weak existence.

Here is a counterexample in dimension one, with $\Theta =1$:Suppose that a solution to the martingale problem with domain $C_c^{\infty }(\mathbb{R})$ existed. The cutoff argument from part (a)(ii), now with $\mathcal{L}=\theta \Delta$, makes $X$ a continuous local martingale withThe criterion for a continuous local martingale with bounded quadratic variation makes it a square-integrable martingale, and the Itô isometry gives $\mathbb{E}{X}_t^2=\mathbb{E}[X{]}_t$.

Apply the occupation-times formula with the indicator of the singleton $\{0\}$. Since that singleton has zero Lebesgue measure,Thus $X$ spends zero ordinary time at zero. Away from zero, $\theta (x)\le x^2$, soThe Gronwall inequality forces $\mathbb{E}{X}_t^2=0$ for every $t$. Continuity then forces $X\equiv 0$ almost surely, contradicting either the zero occupation time just proved or $[X{]}_t=2t$ for that constant path. Consequently

For completeness, the intended construction works if $\theta$ is a Borel measurable function and bounded away from zero on every compact set. In particular, it works for continuous strictly positive $\theta$. Let $B$ be standard $d$-dimensional Brownian motion, and define a clock and its inverse byThe Brownian path on each finite interval has compact range, so the added hypothesis makes $A_u$ finite. The clock is continuous and strictly increasing, and $A_u\ge u/(2\Theta )$, so $A_u\to \infty$. Its inverse is continuous and satisfies ${\tau }_t\le 2\Theta t$. Each ${\tau }_t$ is a stopping time for the Brownian filtration.

For $f\in C_c^{\infty }({\mathbb{R}}^d)$, the optional sampling theorem applied at these bounded random times shows thatis a true martingale in the filtration ${\mathcal{G}}_t={\mathcal{F}}_{{\tau }_t}^B$. The inverse-clock substitution $d{\tau }_t=2\theta (X_t)dt$, valid almost everywhere, rewrites it asThis proves the Brownian time change for a variable diffusion speed:The factor $2$ in the clock is necessary because standard Brownian motion has infinitesimal generator $\Delta /2$.

A sufficient condition is that $x\mapsto \sqrt{2\theta (x)}$ is globally Lipschitz. A more familiar, stronger condition implying this is that $\theta$ is globally Lipschitz and $\theta \ge \epsilon >0$. Both conditions repair the existence issue in part (i).

To connect this to the suggested stochastic differential equation, apply the coordinate and coordinate-product cutoff tests to any solution of the martingale problem. As before they giveDefine, using stochastic integrals against the coordinates of $X$,These integrals exist even without a global positive lower bound: their quadratic variations are exactly $t$. Indeed,The Lévy characterization of multidimensional Brownian motion identifies $W$ as standard $d$-dimensional Brownian motion. By associativity of stochastic integration,The diffusion matrix $\sqrt{2\theta (x)}I$ is globally Lipschitz under the stated condition, and boundedness of $\theta$ gives bounded growth. The global existence theorem for stochastic differential equations with Lipschitz coefficients gives a unique strong solution of a stochastic differential equation and uniqueness in law. Every solution to the martingale problem has just been represented as a weak solution of a stochastic differential equation of this form, so they all have the same law. Conversely, the Itô formula applied to any $f\in C_c^{\infty }$ gives a true martingale, since its stochastic integrand is bounded. Thus the martingale problem is well posed under this sufficient condition.