The Wiener process, the stochastic integral over the Wiener process, and stochastic differential equations

Abstract

Let (Ω, ℱ, P) be a probability space and β = (β _t ), t ≥ 0, be a Brownian motion process (in the sense of the definition given in Section 1.4). Denote ℱ ^β _t = σ{ω: β _s , s ≤ t}. Then, according to (1.30) and (1.31), (P-a.s.)

$$ M({\beta _t}|\mathcal{F}_s^\beta ) = {\beta _s},{\text{ t}} \geqslant {\text{s}} $$

(4.1)

,

$$ M[{({\beta _t} - {\beta _s})^2}|\mathcal{F}_s^\beta ] = t - s,{\text{ t}} \geqslant s $$

(4.2)

.