Stochastic differential equations

Abstract

Consider a complete filtration ${\mathcal{F}}_t$, t∈[0,T]} and an m-dimensional Wiener process {W(t), t ∈ [0,T]} with respect to it. By definition, a stochastic differential equation (SDE) is an equation of the form with X ₀=ξ, where ξ is an ${\mathcal{F}}_0$-measurable random vector, $b=b(t,x): [0,T]\times \mathbb{R}^n\rightarrow \mathbb{R}^n$, and $\sigma=\sigma(t,x): [0,T]\times \mathbb{R}^n\rightarrow \mathbb{R}^{n\times m}$ are measurable functions.