Stochastic differential equations

Abstract

We are given two positive integers r, d and an r-dimensional Brownian motion B(t), t ≥ 0, in a probability space (Ω, ℱ, ℙ) whose natural filtration we denote by (ℱ _t ) _t ≥ 0. We are concerned with the following integral equation,

$$X\left( t \right) = \eta + \int_s^t {b\left( {r,X\left( r \right)} \right)dr} + \int_s^t {\sigma \left( {r,X\left( r \right)} \right)d\,B\left( r \right)} ,$$

(8.1)

where T > 0, s ∈ [0, T), η ∈ L ²(Ω, ℱ _s , ℙ; ℝ^d), b: [0, T] × ℝ^d → ℝ^d and σ : [0, T] × ℝ^d → L(ℝ^r, ℝ^d). b is called the drift and σ the diffusion coefficient of the equation.