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When solving a SDE by Ito's formula, we have to find a function $f(t, X_t)$ of index $t$ and the process $X$ to be solved for.

  1. I was wondering what is the criterion of choosing $f$? Is it to make the integrands not contain $X$?
  2. How is $f$ chosen in principle?

Thanks and regards!

saz
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Tim
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1 Answers1

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It depends on the particular SDE. There is no systematic approach.

One main idea is striving for some cancelation to simplify the drift/volatility coefficients (eg.https://math.stackexchange.com/questions/823785/stochastic-differential-equation-with-trigonometric-functions). For example, in Geometric Brownian motion

$$ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$$

one observes the $S_{t}$ term would cancel out with the derivative of $f(x)=log(x)$.

Another idea is to help transfer some ODE techniques such as integration-factor see here. Solution to General Linear SDE

Thomas Kojar
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