Itô's lemma-Cedar Herald Index

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.

Kiyoshi Itô published a proof of the formula in 1951.

Itô's lemma

Motivation

Suppose we are given the stochastic differential equation

where 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

Itô's lemma

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:

Itô's lemma

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.

Informal derivation

A formal proof of the lemma relies on taking the limit of a sequence of random variables. This approach is not presented here since it involves a number of technical details. Instead, we give a sketch of how one can derive Itô's lemma by expanding a Taylor series and applying the rules of stochastic calculus.

Suppose is an Itô drift-diffusion process that satisfies the stochastic differential equation

where is a Wiener process.

If is a twice-differentiable scalar function, its expansion in a Taylor series is

Substituting for and therefore for gives

In the limit , the terms and tend to zero faster than , which is . Setting the and terms to zero, substituting for (due to the quadratic variation of a Wiener process), and collecting the and terms, we obtain

as required.

Geometric intuition

Suppose we know that are two jointly-gaussian distributed random variables, and is nonlinear but has continuous second derivative, then in general, neither of is gaussian, and their joint distribution is also not gaussian. However, since is gaussian, we might still find is gaussian. This is not true when is finite, but when becomes infinitesimal, this becomes true.

The key idea is that has a deterministic part and a noisy part. When is nonlinear, the noisy part has a deterministic contribution. If is convex, then the deterministic contribution is positive (by Jensen's inequality).

To find out how large is the contribution, we write , where is a standard gaussian, then perform Taylor expansion. We have split it into two parts, a deterministic part, and a random part with mean zero. The random part is non-gaussian, but the non-gaussian parts decay faster than the gaussian part, and at the limit, only the gaussian part remains. The deterministic part has the expected , but also a part contributed by the convexity: .

To understand why there should be a contribution due to convexity, consider the simplest case of geometric Brownian walk (of the stock market): . In other words, . Let , then , and is a Brownian walk. However, although the expectation of remains constant, the expectation of grows. Intuitively it is because the downside is limited at zero, but the upside is unlimited. That is, while is normally distributed, is log-normally distributed.