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 \(F_t^\beta= \sigma \left\{ {\omega :{\beta _s}} \right.,s \leqslant \left. t \right\}\) Then, according to (1.30) and (1.31),(P-a.s) $$M\left( {{\beta _t}|F_s^\beta } \right) = {\beta _s},t \geqslant s $$ (4.1) $$M\left[ {{{\left( {{\beta _t} - {\beta _s}} \right)}^2}|F_s^\beta } \right] = t - s,t \geqslant s. $$ (4.2)