On the Ayed-Kuo stochastic integration for anticipating integrands

Jornet, Marc

doi:10.1080/07362994.2022.2104730

Stochastic Analysis and Applications

2022

DOI: 10.1080/07362994.2022.2104730

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On the Ayed-Kuo stochastic integration for anticipating integrands

Marc Jornet

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2023

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Cited by 2 publications

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A Stratonovich integral for anticipating processes

Jornet

2023

Math Methods in App Sciences

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A stochastic integral for anticipating integrands was introduced by Ayed and Kuo in 2008. Riemann–Stieltjes sums were considered, where the adapted part of the integrand was evaluated at the left endpoints of the subintervals, while the instantly independent part was evaluated at the right endpoints. Since then, many results have been proved, such as formulas for differentials. In this paper, the Stratonovich counterpart of the Ayed–Kuo integral is investigated. In its simplest version, it is proved that, analogously to the classical stochastic integration theory for adapted processes, the fundamental theorem of calculus holds. Consequences, extensions, and limitations are discussed in detail.

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A Stratonovich integral for anticipating processes

Jornet

2023

Math Methods in App Sciences

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Derivative of Certain Stochastic Integrals with Anticipating Integrands

Jornet

2023

Mediterr. J. Math.

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We 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.

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On the Ayed-Kuo stochastic integration for anticipating integrands

Cited by 2 publications

References 21 publications

A Stratonovich integral for anticipating processes

A Stratonovich integral for anticipating processes

Derivative of Certain Stochastic Integrals with Anticipating Integrands

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