# $d$-Dimensional Brownian Motion Martingales

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.

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@ user : Use Itô's lemma and look at the drift part and check that the diffusion part is ok for your process to be a martingale and not only a local martingale. Best regards –  TheBridge Mar 30 '12 at 13:19

On the contrary, these are classic examples of local martingales that are not martingales.

1. Exercise 2.13 (An important counterexample) on page 194 of Continuous Martingales and Brownian Motion (3rd edition) by Daniel Revuz and Marc Yor.

2. Exercises 3.36 and 3.37 on page 168 of Brownian Motion and Stochastic Calculus (2nd edition) by Ioannis Karatzas and Steven E. Shreve.

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