STOCHASTIC INTEGRATION

Abstract

The Wiener integral is the origin of stochastic integrals. The Wiener integral can be generalized in two directions: (1) the stochastic integral and (2) the multiple Wiener integral. If B = (B(t,ω), 0 ≤ t ≤ 1, ω ∈ (Ω,F,P)) is a Brownian motion with B(0,ω) ≡ 0, then B(t,ω) is continuous but nowhere differentiable in t for almost every ω. The Wiener integral can be defined in the same way as the spectral decomposition in the Hilbert space theory. In the Wiener integral, f(t) does not depend on ω but only on t, that is, f(t) is a deterministic function in probability language.