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Applying Itô’s Lemma to Transform Stochastic Differential Equations

Article Quant Q&A · Author: Sandro
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Summary

The document shows how to derive the dynamics of a transformed stochastic process from the dynamics of its underlying process. For a geometric diffusion and a power transformation, Itô’s lemma produces a new equation with both drift and stochastic terms; the resulting coefficients help identify the distributional form. The response points to a lognormal process as the expected form for the first transformation discussed.

For a mean-reverting process with proportional diffusion, the example applies the second-order term in Itô’s lemma to the transformation Y = S². It derives a drift containing both the original mean-reversion component and a variance correction, alongside a stochastic term proportional to the transformed state. The document gives an algebraic derivation, not empirical evidence. Its first question is ambiguous and the brief suggested result may depend on how the transformation is interpreted, so careful specification of the function and notation is needed before applying the method.

Key ideas

  • Itô’s lemma converts the dynamics of a process into the dynamics of a smooth function of that process.
  • The transformed equation generally contains both drift and stochastic components.
  • The second derivative term contributes a variance correction to the drift.
  • For a squared mean-reverting process with proportional diffusion, the transformed drift depends on both the state and its square root.

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# Stochastic process theory question


# Stochastic process theory question












*S follows a process $dS= mSdt + oSdz$ where m and o are constant.

What is the probability followed by $ Y=(Se)^{(r-t)} $.

If S follows a process $ dS= k (b-S) dt + oSdz $ where k, b, o are constant.

What’s the process followed by $Y =S^2$ ?

## Answer by Uditg_ucla (score 1)

https://quant.stackexchange.com/a/22184

Not sure I fully understand your question. However, I'd suggest using the Ito's lemma (second equation on wikipedia page https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma) to solve for dY. In both cases, dY will have both a drift term and a stochastic term. The coefficient of the stochastic term will indicate what sort of probability process Y follows.

e.g., in the first case, you'll get something like dY = [ (m-1)Y ]dt + [ rY ]dW, implying log-normal distribution for Y.

## Answer by Neeraj (score 1)

https://quant.stackexchange.com/a/23161

The first part has already been answer by @Uditg_ucla, so I am only providing answer of your 2nd part.

Rewriting your SDE in more sophisticated way: $$dS=k(b-S)dt+\sigma S dz$$ You want SDE for $S^2$. Using Taylor series, it can be written as: $$df(S)=f'(S)dS + \frac{1}{2!}f''(S)(dS)^2+\cdots$$ $$df(S)=2SdS+(dS)^2$$ $$df(S)=2S[k(b-S)dt+\sigma S dz]+\sigma^2 S^2 dt$$ $$df(S)=\bigg(2Sk(b-S)+\sigma^2S^2\bigg)dt+2\sigma S^2dz$$ since $Y=S^2$, so replacing $S^2$ from $Y$, $$dY=\bigg(2k(b\sqrt{Y}-Y)+\sigma^2Y\bigg)dt+2\sigma Y dz$$ desired SDE...

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.