Transforming a Diffusion to Constant Volatility with Itô’s Lemma
Summary
The document explains how to transform a one-dimensional diffusion into a process with constant diffusion coefficient. Applying Itô’s lemma to a function of the state shows that the stochastic term is the derivative of the function multiplied by the original volatility. Equating that term to a constant gives a first-order differential equation for the transformation.
Integrating the reciprocal of the state-dependent volatility yields the family of valid transformations. The additive constant is arbitrary, and if the target constant volatility is not fixed, its scale is arbitrary as well. The drift of the transformed process follows from the remaining Itô terms. The construction assumes positive volatility and sufficient regularity for Itô’s lemma and the integral to apply; it does not address boundary behavior or whether the transformed process has any particular financial interpretation.
Key ideas
- Itô’s lemma gives the transformed diffusion coefficient as the function’s derivative times the original volatility.
- A constant target diffusion requires the derivative to be proportional to the reciprocal of the original volatility.
- Integrating that derivative determines the transformation up to an additive constant.
- If the target volatility is unspecified, the transformation also has an arbitrary scale.
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# Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance
# Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance
> If $$dX_t = \mu(t,X_t)dt + \sigma(X_t)dW_t$$ with $\sigma$ positive, show there exists a function $f$ such that $$d\left(f(X_t)\right) = v(t,X_t)dt + V dW_t$$ where $V$ is constant. How unique is $f$?
The solution by Mark Joshi says: The volatility of $f(X_t)$ will, from Ito's lemma, be $$f'(X_t)\sigma(X_t)$$ So we need to solve $$f'(X_t) = \sigma^{-1}(X_t)A$$ for some constant $A$. We deduce $$f(X) = C + A\int_{0}^{X}\sigma(S)^{-1}dS$$ with $A$ and $C$ arbitrary constants.
Is this a complete solution? I don't see the associated steps in showing if $f$ is unique. Also, I do not understand why when he asserts from Ito's lemma that the volatility part of $f(X_t)$ leads to solving $f'(X_t) = \sigma^{-1}(X_t)A$.
## Answer by Quantuple (score 4, accepted)
https://quant.stackexchange.com/a/37598
It is a complete solution.
Bearing in mind the SDE verified by $(X_t)_{t \geq 0}$, applying Itô's lemma to compute the (stochastic) differential of $f(X_t)$ yields \begin{align} df(X_t) &= \underbrace{\frac{\partial f}{\partial t}}_{0} dt + \frac{\partial f}{\partial X}(X_t) dX_t + \frac{1}{2} \frac{\partial^2 f}{\partial X^2}(X_t) d\langle X \rangle_t \\ &= f'(X_t) dX_t + \frac{1}{2}f''(X_t)\sigma^2(X_t) dt \\ &= \left( f'(X_t)\mu(t,X_t) + \frac{1}{2}f''(X_t)\sigma^2(X_t) \right) dt +f'(X_t) \sigma(X_t) dW_t \\ &:= v(t,X_t) dt + V dW_t \end{align} such that by identifiying the diffusion terms we indeed need to solve $$ f'(X_t)\sigma(X_t) = V $$ Solving this simple ODE gives $$ f(X) = C + V \int_0^X \sigma^{-1}(x) dx,\,\,\, \forall C \in \Bbb{R} $$ which shows the function $f$ is not unique since defined up to a scalar constant (obviously if $V$ is not fixed then it is also an arbitrary parameter).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.