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Concept
Itô's lemma
Summary
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in the French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values.
Suppose we are given the stochastic differential equation
where Bt is a Wiener process and the functions are deterministic (not stochastic) functions of time. In general, it's not possible to write a solution directly in terms of However, we can formally write an integral solution
This expression lets us easily read off the mean and variance of (which has no higher moments). First, notice that every individually has mean 0, so the expectation value of is simply the integral of the drift function:
Similarly, because the terms have variance 1 and no correlation with one another, the variance of is simply the integral of the variance of each infinitesimal step in the random walk:
However, sometimes we are faced with a stochastic differential equation for a more complex process in which the process appears on both sides of the differential equation. That is, say
for some functions and In this case, we cannot immediately write a formal solution as we did for the simpler case above. Instead, we hope to write the process as a function of a simpler process taking the form above. That is, we want to identify three functions and such that and In practice, Ito's lemma is used in order to find this transformation. Finally, once we have transformed the problem into the simpler type of problem, we can determine the mean and higher moments of the process.
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