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Randomization in the first hitting time problem

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  • Jackson, Ken
  • Kreinin, Alexander
  • Zhang, Wanhe

Abstract

In this paper, we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F(t), and a linear boundary, b(t)=[mu]t, find a distribution of the initial state such that the distribution of the first hitting time is F(t). This problem has important applications in credit risk modeling where the process represents the so-called distance to default of an obligor, the first hitting time represents a default event and the boundary separates the healthy states of the obligor from the default state. We show that randomization of the initial state of the process makes the problem analytically tractable.

Suggested Citation

  • Jackson, Ken & Kreinin, Alexander & Zhang, Wanhe, 2009. "Randomization in the first hitting time problem," Statistics & Probability Letters, Elsevier, vol. 79(23), pages 2422-2428, December.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:23:p:2422-2428
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    References listed on IDEAS

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    1. Duffie, Darrell & Lando, David, 2001. "Term Structures of Credit Spreads with Incomplete Accounting Information," Econometrica, Econometric Society, vol. 69(3), pages 633-664, May.
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    Cited by:

    1. Abundo, Mario, 2012. "An inverse first-passage problem for one-dimensional diffusions with random starting point," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 7-14.
    2. Zhang, Ao & Wang, Zhihua & Bao, Rui & Liu, Chengrui & Wu, Qiong & Cao, Shihao, 2023. "A novel failure time estimation method for degradation analysis based on general nonlinear Wiener processes," Reliability Engineering and System Safety, Elsevier, vol. 230(C).
    3. Mario Abundo, 2018. "The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion," Mathematics, MDPI, vol. 6(6), pages 1-10, May.
    4. Alex Langnau & Yanko Punchev, 2011. "Stochastic Price Dynamics Implied By the Limit Order Book," Papers 1105.4789, arXiv.org.
    5. Abundo, Mario, 2013. "The double-barrier inverse first-passage problem for Wiener process with random starting point," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 168-176.
    6. Andrea Valsecchi & Leonardo Vanneschi & Giancarlo Mauri, 2014. "A study of search algorithms’ optimization speed," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 256-270, February.
    7. Gorno, Leandro & Iachan, Felipe S., 2020. "Competitive real options under private information," Journal of Economic Theory, Elsevier, vol. 185(C).
    8. Renault, Eric & van der Heijden, Thijs & Werker, Bas J.M., 2014. "The dynamic mixed hitting-time model for multiple transaction prices and times," Journal of Econometrics, Elsevier, vol. 180(2), pages 233-250.

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