Let $d > 1$ and let $W_t$ denote a standard $d$-dimensional Brownian motion starting at $x\neq 0$. Let $M_t = \log|W_t|$ for $d = 2$, and $M_t= |W_t|^{2-d}$ for $d > 2$. Show that $M_t$ is a martingale.
2026-04-05 19:02:23.1775415743
$d$-Dimensional Brownian Motion Martingales
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On the contrary, these are classic examples of local martingales that are not martingales.
Exercise 2.13 (An important counterexample) on page 194 of Continuous Martingales and Brownian Motion (3rd edition) by Daniel Revuz and Marc Yor.
Exercises 3.36 and 3.37 on page 168 of Brownian Motion and Stochastic Calculus (2nd edition) by Ioannis Karatzas and Steven E. Shreve.