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Kernel representation formula: From complex to real Wiener–Itô integrals and vice versa

Author

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  • Chen, Huiping
  • Chen, Yong
  • Liu, Yong

Abstract

We characterize the relation between the real and complex Wiener–Itô integrals. Given a complex multiple Wiener–Itô integral, we get explicit expressions for the kernels of its real and imaginary parts. Conversely, considering a two-dimensional real Wiener–Itô integral, we obtain the representation formula by a finite sum of complex Wiener–Itô integrals. The main tools are a recursion technique and Malliavin derivative operators. As an application to stochastic processes, we investigate the regularity of the stationary solution of the stochastic heat equation with dispersion.

Suggested Citation

  • Chen, Huiping & Chen, Yong & Liu, Yong, 2024. "Kernel representation formula: From complex to real Wiener–Itô integrals and vice versa," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    • Handle: RePEc:eee:spapps:v:167:y:2024:i:c:s0304414923002132
    DOI: 10.1016/j.spa.2023.104241
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    References listed on IDEAS

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    1. Marinucci, Domenico & Peccati, Giovanni, 2008. "High-frequency asymptotics for subordinated stationary fields on an Abelian compact group," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 585-613, April.
    2. Pham, Viet-Hung, 2013. "On the rate of convergence for central limit theorems of sojourn times of Gaussian fields," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2158-2174.
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