= Brownian time change for a variable diffusion speed
{c}
{title2=$X_t=B_{A^{-1}(t)},\quad A_u=\int_0^u[2\theta(B_r)]^{-1}dr$}
Let $\theta:\mathbb R^d\to(0,\Theta]$ be a <Borel measurable function> bounded away from zero on every compact set. If $B$ is standard $d$-dimensional <Brownian motion>, the displayed clock is finite on compact intervals, continuous, strictly increasing, and unbounded. Its inverse $\tau_t=A^{-1}(t)$ is a Brownian <stopping time>, with $\tau_t\leq2\Theta t$. The <optional sampling theorem> and the change of variables $d\tau_t=2\theta(B_{\tau_t})dt$ show that $X_t=B_{\tau_t}$ solves the <martingale problem> for $\mathcal L=\theta\Delta$ on <test functions> $C_c^\infty(\mathbb R^d)$. The <filtration> is $\mathcal F^B_{\tau_t}$. These are true <martingales>: on a finite new-time horizon the sampling times have a deterministic bound. Continuous strictly positive $\theta$ satisfies the compact-set lower bound. Pointwise positivity alone does not; <pointwise positivity does not ensure weak existence> gives a counterexample.
Back to article page
