Itô process

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A stochastic process with a stochastic differential. More precisely, a continuous stochastic process $X_t$ on a probability space $(\Omega, \mathcal{F}, \Prob)$ with a certain non-decreasing family $\{\mathcal F_t\}$ of $\sigma$-algebras of is called an Itô process with respect to if there exists processes and (called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to for each , and a Wiener process with respect to , such that Such processes are called after K. Itô , . One and the same process can be an Itô process with respect to two different families . The corresponding stochastic differentials may differ substantially in this case. An Itô process is called a process of diffusion type (cf. also Diffusion process) if its drift coefficient and diffusion coefficient are, for each , measurable with respect to the -algebra Under certain, sufficiently general, conditions it is possible to represent an Itô process as a process of diffusion type, but, generally, with some new Wiener process (cf. ). If an Itô process is representable as a diffusion Itô process with some Wiener process and if the equation is satisfied, then is called the innovation process for .

Examples. Suppose that a certain Wiener process , , with respect to has been given and suppose that where is a normally-distributed random variable with mean and variance that is measurable with respect to .

The process , regarded with respect to , has stochastic differential in which the new Wiener process , defined by is an innovation process for .