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An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems

Author

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  • Da Pelo, Paolo
  • Lanconelli, Alberto
  • Stan, Aurel I.

Abstract

The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type.

Suggested Citation

  • Da Pelo, Paolo & Lanconelli, Alberto & Stan, Aurel I., 2013. "An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3183-3200.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:8:p:3183-3200
    DOI: 10.1016/j.spa.2013.03.005
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    Citations

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    Cited by:

    1. Bilel Kacem Ben Ammou & Alberto Lanconelli, 2019. "Rate of Convergence for Wong–Zakai-Type Approximations of Itô Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1780-1803, December.
    2. Kang, Yuanbao & Wang, Caishi, 2014. "Itô formula for one-dimensional continuous-time quantum random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 414(C), pages 154-162.
    3. Lanconelli, Alberto, 2018. "Standardizing densities on Gaussian spaces," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 243-250.
    4. Lanconelli, Alberto & Scorolli, Ramiro, 2021. "Wong–Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 57-78.

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