随机积分和随机方程.pdf

Intro Brownian motion Integration SDE Applications Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 2007 Oleg Makhnin New Mexico Tech Dept. of Mathematics Stochastic Integration and Stochastic Differential Equations: a gentle introduction Intro Brownian motion Integration SDE Applications Intro: why Stochastic? Brownian Motion/ Wiener process Stochastic Integration Stochastic DE’s Applications Oleg Makhnin New Mexico Tech Dept. of Mathematics Stochastic Integration and Stochastic Differential Equations: a gentle introduction Intro Brownian motion Integration SDE Applications Intro Deterministic ODE X (t) = a(t, X (t)) t 0 X (0) = X0 However, “noise” is usually present X (t) = a(t, X (t)) + b(t, X (t)) ω (t), t 0 X (0) = X0 Natural requirements for noise ω : - ω (t) is random with mean 0 - ω (t) is independent of ω (s ), t = s ⇒ “White noise” - ω is continuous Strictly speaking, ω does not exist! Oleg Makhnin New Mexico Tech Dept. of Mathematics Stochastic Integration and Stochastic Differential Equations: a gentle introduction Intro Brownian motion Integration SDE Applications Brownian motion