Itô calculus
Itô calculus, named after Kiyosi Itô, extends a methods of calculus to stochastic processes such(a) as Brownian motion see Wiener process. It has important applications in mathematical finance in addition to stochastic differential equations.
The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands as well as the integrators are now stochastic processes:
where H is a locally square-integrable process adapted to the Revuz & Yor 1999, Chapter IV, which is a Brownian motion or, more generally, a semimartingale. The written of the integration is then another stochastic process. Concretely, the integral from 0 to any particular t is a random variable, defined as a limit of asequence of random variables. The paths of Brownian motion fail to satisfy the specifics to be professionals to apply the standards techniques of calculus. So with the integrand a stochastic process, the Itô stochastic integral amounts to an integral with respect to a function which is not differentiable at any member and has infinite variation over every time interval. The leading insight is that the integral can be defined as long as the integrand H is adapted, which generally speaking means that its good at time t can only depend on information usable up until this time. Roughly speaking, one chooses a sequence of partitions of the interval from 0 to t and make-up Riemann sums. Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. it is for crucial which constituent in regarded and talked separately. of the small intervals is used to compute the return of the function. The limit then is taken in probability as the mesh of the partition is going to zero. numerous technical details clear to be taken care of to show that this limit exists and is freelancer of the particular sequence of partitions. Typically, the left end of the interval is used.
Important results of Itô calculus include the integration by parts formula and Itô's lemma, which is a change of variables formula. These differ from the formulas of standard calculus, due to quadratic variation terms.
In Revuz & Yor 1999, Chapter IV.