Stochastic Integral

k=1 Gk(ω) · 1(uk,vk](s), ω ∈ Ω, s ≥ 0, where n ∈ N, 0 ≤ uk < vk, and Gk ∈ L(Fuk). We let Le denote the class of all such integrands. Notice that each H ∈ Le satisfies (i) s → Hs(ω) is a left-continuous step function for each ω ∈ Ω, (ii) Hs ∈ L(Fs) for each s ≥ 0, (iii) viewed as a mapping from Ω × [0, t] to R, (ω, s) → Hs(ω) is Ft ⊗B[0,t] measurable for each t > 0, and (iv) E[ ∫ t 0 H s ds] < ∞ for each t > 0. We now define, for H ∈ Le as displayed in (2.1),