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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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Related concepts (9)
Stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces.
Itô calculus
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: where H is a locally square-integrable process adapted to the filtration generated by X , which is a Brownian motion or, more generally, a semimartingale.
Stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale.
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