# Transformation of Ito diffusion such that the diffusion coefficient becomes constant (Change of variables)

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 $$f$$?

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.