Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
@ 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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.