Ito process

An Ito Process is a type of stochastic process described by Japanese mathematician Kiyoshi Ito, which can be written as the sum of the integral of a process over time and of another process over a Brownian Motion.

Those processes are the base of Stochastic Integration, and are therefore widely used in Financial Mathematics and Stochastic Calculus.

Description of the Ito Processes

Let $(\Omega ,F,\mathbb {F} ,\mathbb {P} )$ be a probability space with a filtration $\mathbb {F} =(F_{t})_{t\geq 0}$ that we consider as complete (that is to say, all sets which measure is null are contained in $F_{0}$

Let also be $B=(B_{t}^{1},\dots ,B_{t}^{d})_{t\geq 0}$ a d-dimensional $\mathbb {F}$- Standard Brownian Motion.

Then we call Ito Process all process $(X_{t})_{t\geq 0}$ that can be written like :

$X_{t}=X_{0}+\int _{0}^{t}K_{s}\mathrm {ds} +\sum _{j=1}^{d}\int _{0}^{t}H_{s}^{j}*\mathrm {dB} _{s}^{j}$

Where :

• $X_{0}$ is $F_{0}$ measurable
• $(K_{t})_{t\geq 0}$ is a progressively measurable process such as $\forall t\geq 0,\ \int _{0}^{t}|K_{s}|{\textrm {ds}}<+\infty$ almost surely.
• $(H_{t}^{i})_{t\geq 0,\ i\in [1\dots d]}$ is progressively measurable and such as $\forall i\in [1\dots d],\ \forall t\geq 0,\ \int _{0}^{t}(H_{s}^{i})^{2}\mathrm {(} ds)<+\infty$ almost surely.