3
$\begingroup$

I encountered a transformation of an Ito diffusion which yielded a new Ito diffusion such that the new diffusion coefficient was equal to $1$.

My question is whether there is something more general behind it and, if yes, whether it can be found anywhere in a book.

EDIT: I found that this is the so-called Lamperti-Transform, see e.g. here

Let me present you the transformation:

Let $\mu : \Bbb R \times [0,\infty ) \to \Bbb R$ and $\sigma : \Bbb R \times [0,\infty ) \to \Bbb R$ be smooth and bounded functions. Let $ (X_t )_{t\geq 0} $ be the solution to

$$ dX_t = \mu (X_t , t ) dt + \sigma (X_t ,t) dB_t $$

where $(B_t)_{t\geq 0}$ is a Brownian motion.

Assume that $$\inf_{(x,t) \in \Bbb R \times [0,\infty)} \sigma (x,t) > 0.$$

Define the function

$$f(x,t) = \int_0^x \frac{1}{\sigma (z,t) } dz $$ We have that $f \in C^{2,1}$ and for fixed $t$ the function $x\mapsto f(x,t)$ is strictly increasing, such that there exists a function $g : \Bbb R \times [0,\infty) \to \Bbb R$ with $$ f(g(y,t),t) = y$$ Using the Ito formula for the process $Y_t := f(X_t,t)$ we obtain that

$$ dY_t = \left[ \frac{\mu (X_t ,t)}{\sigma (X_t , t)} - \int_0^{X_t} \frac{\frac{\partial\sigma}{\partial t} (z, t)}{\sigma (z,t)^2}dz - \frac 1 2 \frac{\partial \sigma}{\partial x} (X_t , t)\right] dt + dB_t\\ = \tilde \mu (Y_t , t) dt + dB_t, $$

where

$$\tilde \mu (y,t) := \frac{\mu (g(y,t) ,t)}{\sigma (g(y,t) , t)} - \int_0^{g(y,t)} \frac{ \frac{\partial\sigma}{\partial t} (z,t)}{\sigma (z,t)^2 }dz - \frac 1 2 \frac{\partial \sigma}{\partial x} (g(y,t) , t) .$$

Therefore, $(Y_t)_{t\geq 0}$ is a Ito diffusion with diffusion coefficient equal to $1$.

  • My question for a possible more general situation is concerned with the condition $\sigma \geq \varepsilon >0$. Maybe it is possible to generalize it to the situation that $\sigma^2 > 0$ and $\sigma^{-1}$ is locally integrable.

  • A different way of transforming the diffusion coefficient is the method of random time change. Is there any relationship to the deterministic transformation with $f$?

$\endgroup$

1 Answer 1

1
$\begingroup$

For Q1, in the references I found for Lamperti

  • The Lamperti Transform
  • Schilling in "21.4 Transforming an SDE into a linear SD"
  • Shreve-Karatzas 5.2 exercise 2.20

they all work with $\sigma^{2}>0$ (continuously differentiable).

For Q2, well the Lampert is about turning the volatility into $1$ (even if the drift is nonzero), whereas the time-change method requires zero drift as 5.5.A in Shreve-Karatzas or Dubins-Schwartz is about obtaining a time-changed Brownian motion, so they are quite different.

$\endgroup$

You must log in to answer this question.