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First exit times of SDEs driven by stable Lévy processes

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  • Imkeller, P.
  • Pavlyukevich, I.

Abstract

We study the exit problem of solutions of the stochastic differential equation from bounded or unbounded intervals which contain the unique asymptotically stable critical point of the deterministic dynamical system . The process L is composed of a standard Brownian motion and a symmetric [alpha]-stable Lévy process. Using probabilistic estimates we show that, in the small noise limit [epsilon]-->0, the exit time of X[epsilon] from an interval is an exponentially distributed random variable and determine its expected value. Due to the heavy-tail nature of the [alpha]-stable component of L, the results differ strongly from the well known case in which the deterministic dynamical system undergoes purely Gaussian perturbations.

Suggested Citation

  • Imkeller, P. & Pavlyukevich, I., 2006. "First exit times of SDEs driven by stable Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 116(4), pages 611-642, April.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:4:p:611-642
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    References listed on IDEAS

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    1. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53, January.
    2. Samorodnitsky, G. & Grigoriu, M., 2003. "Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Lévy motions," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 69-97, May.
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    Cited by:

    1. Jakubowski, Tomasz, 2007. "The estimates of the mean first exit time from a ball for the [alpha]-stable Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1540-1560, October.
    2. Tong, Changqing & Lin, Zhengyan & Zheng, Jing, 2012. "The local time of the Markov processes of Ornstein–Uhlenbeck type," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1229-1234.
    3. Pavlyukevich, Ilya, 2008. "Simulated annealing for Lévy-driven jump-diffusions," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 1071-1105, June.
    4. Valentin Konakov & Stéphane Menozzi, 2011. "Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities," Journal of Theoretical Probability, Springer, vol. 24(2), pages 454-478, June.
    5. Wang, Xiao & Duan, Jinqiao & Li, Xiaofan & Luan, Yuanchao, 2015. "Numerical methods for the mean exit time and escape probability of two-dimensional stochastic dynamical systems with non-Gaussian noises," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 282-295.
    6. Gairing, Jan & Högele, Michael & Kosenkova, Tetiana, 2018. "Transportation distances and noise sensitivity of multiplicative Lévy SDE with applications," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2153-2178.

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