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Appendix 17.1: Derivation of Ito's Lemma

Let X be a generalized Wiener process whose law of motion is described by:

Now, define f(x) as a real valued function of X—for example, f(x) is a derivative of X whose law of motion we wish to determine. Expand f(x) using a Taylor series:

Then, in the limit, as we get:

We are recognizing here that this result holds because the higher order terms go to zero faster as Δ approaches zero. This is the fundamental theorem of calculus. But in stochastic calculus, the second order term does not vanish because X is normally distributed with positive variance, which converges in probability to . This can be conceptualized from the Wiener process itself, where the term has variance , since .

So, while the fundamental theorem of calculus is:

we must extend ...