# Lamperti Transform

Let $$X_t$$ be an Ito process given by
$$dX_t=f(X_t,t)dt+\sigma(X_t,t)dw_t$$

The Lamperti transformation is, $$Z_t=\phi(X_t,t)=\int\frac{1}{\sigma(x,t)}dx$$

and has a unit diffusion; that is, $$dZ_t=(…)dt + dw_t$$.

This transformation is fairly well used in finance, and particularly useful with separable volatility functions both in analytic and numerical contexts. However I am not sure the name is well known and I have to confess I was not aware of the term Lamperti Transform until it was pointed out to me very recently, so this short note is as much for myself as it is for anyone else. I am glad it has a name, it can make it easier to communicate the idea by email in technical discussions.

The paper I read recently explicitly referring to the Lamperti transformation is “From State Dependent Diffusion to Constant Diffusion in Stochastic Differential Equations by the Lamperti Transform”, by Jan Kloppenborg Møller and Henrik Madsen, December 14, 2010.

John Lamperti published a series of papers in the 60’s relating to stochastic processes.

Stay informed with our FREE newsletter, subscribe here.