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Large deviations for multi-scale jump-diffusion processes

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  • Kumar, Rohini
  • Popovic, Lea

Abstract

We obtain large deviation results for a two time-scale model of jump-diffusion processes. The processes on the two time scales are fully inter-dependent, the slow process has small perturbative noise and the fast process is ergodic. Our results extend previous large deviation results for diffusions. We provide concrete examples in their applications to finance and biology, with an explicit calculation of the large deviation rate function.

Suggested Citation

  • Kumar, Rohini & Popovic, Lea, 2017. "Large deviations for multi-scale jump-diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1297-1320.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:4:p:1297-1320
    DOI: 10.1016/j.spa.2016.07.016
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    References listed on IDEAS

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    1. Dupuis, Paul & Spiliopoulos, Konstantinos, 2012. "Large deviations for multiscale diffusion via weak convergence methods," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1947-1987.
    2. Jin Feng & Jean-Pierre Fouque & Rohini Kumar, 2010. "Small-time asymptotics for fast mean-reverting stochastic volatility models," Papers 1009.2782, arXiv.org, revised Aug 2012.
    3. Chen, Zhen-Qing & Wang, Jian, 2014. "Ergodicity for time-changed symmetric stable processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2799-2823.
    4. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    5. Veretennikov, A. Yu., 2000. "On large deviations for SDEs with small diffusion and averaging," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 69-79, September.
    6. Budhiraja, Amarjit & Chen, Jiang & Dupuis, Paul, 2013. "Large deviations for stochastic partial differential equations driven by a Poisson random measure," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 523-560.
    7. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    Cited by:

    1. Jian Wang, 2019. "Compactness and Density Estimates for Weighted Fractional Heat Semigroups," Journal of Theoretical Probability, Springer, vol. 32(4), pages 2066-2087, December.
    2. Gajda, J. & Wyłomańska, A. & Kantz, H. & Chechkin, A.V. & Sikora, G., 2018. "Large deviations of time-averaged statistics for Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 143(C), pages 47-55.
    3. Popovic, Lea, 2019. "Large deviations of Markov chains with multiple time-scales," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3319-3359.

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