Section 18.5

Martingales from Ito's Lemma

Finding the drift correction or constant that makes a process a martingale.

Toolkit

Apply Itô’s lemma to $f(t,W_t)$ :

df = \left(f_t + \tfrac12 f_{ww}\right)dt + f_w\,dW_t.

The process is a (local) martingale iff the $dt$ -term vanishes identically:

f_t(t,w) + \tfrac12 f_{ww}(t,w) = 0,

i.e. $f$ solves the backward heat equation. For more general diffusions $dX_t=\mu\,dt+\sigma\,dW_t$ , the drift condition is

f_t + \mu f_x + \tfrac12\sigma^2 f_{xx} = 0.

Use this to pin down unknown constants/exponents in a trial function (e.g. $f=e^{\alpha W_t+\beta t}$ , $f=W_t^n+\dots$ ) by forcing the coefficient of $dt$ to be zero, or to check a given $f$ is harmonic for the generator. Boundary/initial conditions then fix any remaining free constants.

Problems

18.12 Killing the Drift+3 variations Easy Free