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The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion

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  • Mario Abundo

    (Dipartimento di Matematica, Università Tor Vergata, 00133 Rome, Italy)

Abstract

Let X ( t ) be a continuously time-changed Brownian motion starting from a random position η , S ( t ) a given continuous, increasing boundary, with S ( 0 ) ≥ 0 , P ( η ≥ S ( 0 ) ) = 1 , and F an assigned distribution function. We study the inverse first-passage time problem for X ( t ) , which consists in finding the distribution of η such that the first-passage time of X ( t ) below S ( t ) has distribution F , generalizing the results, valid in the case when S ( t ) is a straight line. Some explicit examples are reported.

Suggested Citation

  • Mario Abundo, 2018. "The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion," Mathematics, MDPI, vol. 6(6), pages 1-10, May.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:6:p:91-:d:149402
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    References listed on IDEAS

    as
    1. Hieber, Peter & Scherer, Matthias, 2012. "A note on first-passage times of continuously time-changed Brownian motion," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 165-172.
    2. T. R. Hurd, 2009. "Credit risk modeling using time-changed Brownian motion," Papers 0904.2376, arXiv.org.
    3. T. R. Hurd, 2009. "Credit Risk Modeling Using Time-Changed Brownian Motion," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(08), pages 1213-1230.
    4. 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.
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