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Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions

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Listed:
  • Jiang Hui

    (Nanjing University of Aeronautics and Astronautics)

  • Xu Lihu

    (University of Macau
    Zhuhai UM Science and Technology Research Institute)

  • Yang Qingshan

    (Northeast Normal University)

Abstract

In this paper, we establish the functional large deviation principle (LDP) for the Kac–Stroock approximations of a wild class of Gaussian processes constructed by telegraph types of integrals with $$L^2$$ L 2 -integrands under mild conditions and find the explicit form for their rate functions. Our investigation includes a broad range of kernels, such as those related to Brownian motions, fractional Brownian motions with whole Hurst parameters, and Ornstein–Uhlenbeck processes. Furthermore, we consider a class of non-Markovian stochastic differential equations driven by the Kac–Stroock approximation and establish their Freidlin–Wentzell type LDP. The rate function clearly indicates an interesting phase transition phenomenon as the approximating rate moves from one region to the other.

Suggested Citation

  • Jiang Hui & Xu Lihu & Yang Qingshan, 2024. "Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3015-3054, November.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01354-0
    DOI: 10.1007/s10959-024-01354-0
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    References listed on IDEAS

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    1. Archil Gulisashvili, 2017. "Large deviation principle for Volterra type fractional stochastic volatility models," Papers 1710.10711, arXiv.org, revised Aug 2018.
    2. Foong, S. K. & Kanno, S., 1994. "Properties of the telegrapher's random process with or without a trap," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 147-173, September.
    3. Bardina, Xavier & Márquez, Juan Pablo & Quer-Sardanyons, Lluís, 2020. "Weak approximation of the complex Brownian sheet from a Lévy sheet and applications to SPDEs," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5735-5767.
    4. Cinque, Fabrizio & Orsingher, Enzo, 2021. "On the exact distributions of the maximum of the asymmetric telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 601-633.
    5. Peter Friz & Stefan Gerhold & Arpad Pinter, 2018. "Option pricing in the moderate deviations regime," Mathematical Finance, Wiley Blackwell, vol. 28(3), pages 962-988, July.
    6. Claudio Macci & Barbara Martinucci & Enrica Pirozzi, 2021. "Asymptotic Results for the Absorption Time of Telegraph Processes with Elastic Boundary at the Origin," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 1077-1096, September.
    7. Antoine Jacquier & Alexandre Pannier, 2020. "Large and moderate deviations for stochastic Volterra systems," Papers 2004.10571, arXiv.org, revised Apr 2022.
    8. Guillin, Arnaud, 2001. "Moderate deviations of inhomogeneous functionals of Markov processes and application to averaging," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 287-313, April.
    9. Bogachev, Leonid & Ratanov, Nikita, 2011. "Occupation time distributions for the telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1816-1844, August.
    10. Orsingher, Enzo, 1990. "Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 49-66, February.
    11. Fontbona, Joaquin & Guérin, Hélène & Malrieu, Florent, 2016. "Long time behavior of telegraph processes under convex potentials," Stochastic Processes and their Applications, Elsevier, vol. 126(10), pages 3077-3101.
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