Derivative of Certain Stochastic Integrals with Anticipating Integrands

Abstract

AbstractWe investigate whether an analogue of the fundamental theorem of calculus holds for the Ayed–Kuo stochastic integral. This integral was defined for anticipating processes $$\phi =\{\phi _t:t\in [a,b]\}$$ ϕ = { ϕ t : t ∈ [ a , b ] } of the form $$\phi _t=h_t\psi _t$$ ϕ t = h t ψ t , where h is adapted and $$\psi $$ ψ is instantly independent with respect to the forward filtration of Brownian motion B. If $$Y_t=\int _a^t \phi _s\text {d}B_s$$ Y t = ∫ a t ϕ s d B s , we aim at proving that $$\begin{aligned} \underset{\Delta t\rightarrow 0}{\text {plim}}\left( \frac{\Delta Y_t}{\Delta B_t}-h_t\psi _{t+\Delta t}\right) =0 \end{aligned}$$ plim Δ t → 0 Δ Y t Δ B t - h t ψ t + Δ t = 0 under different scenarios, where the limit is taken to be in probability.