Stochastic calculus

As a stochastic analysis, the branch of mathematics is called, which deals with the properties of stochastic integrals. Here, a random dependent integral is defined by a stochastic process, ie a random function as an integrator (and often as integrand ) is selected. The value of the integral is therefore a random variable.

As an integrator, a Wiener process is often used, although more general classes of stochastic processes ( Levy processes or more general semimartingales ) are possible. Among the various definitions of the integral Itō integral is the most common, because here is a martingale under certain conditions defined by the process.

An important aspect of stochastic analysis is the formulation and analysis of stochastic differential equations, in particular with regard to the question of the solvability of such equations. This area is closely related to the classical theory of partial differential equations, in part, can be deterministic equations solved using stochastic methods.

Applications can be found in the modeling of processes in financial mathematics, but also in biology and physics. A prominent example is the modeling of stock price developments can be studied with the help of stochastic differential equations.