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On the First-Passage Time Problem for a Feller-Type Diffusion Process

Author

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  • Virginia Giorno

    (Dipartimento di Informatica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano, Salerno, Italy
    These authors contributed equally to this work.)

  • Amelia G. Nobile

    (Dipartimento di Informatica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano, Salerno, Italy
    These authors contributed equally to this work.)

Abstract

We consider the first-passage time problem for the Feller-type diffusion process, having infinitesimal drift B 1 ( x , t ) = α ( t ) x + β ( t ) and infinitesimal variance B 2 ( x , t ) = 2 r ( t ) x , defined in the space state [ 0 , + ∞ ) , with α ( t ) ∈ R , β ( t ) > 0 , r ( t ) > 0 continuous functions. For the time-homogeneous case, some relations between the first-passage time densities of the Feller process and of the Wiener and the Ornstein–Uhlenbeck processes are discussed. The asymptotic behavior of the first-passage time density through a time-dependent boundary is analyzed for an asymptotically constant boundary and for an asymptotically periodic boundary. Furthermore, when β ( t ) = ξ r ( t ) , with ξ > 0 , we discuss the asymptotic behavior of the first-passage density and we obtain some closed-form results for special time-varying boundaries.

Suggested Citation

  • Virginia Giorno & Amelia G. Nobile, 2021. "On the First-Passage Time Problem for a Feller-Type Diffusion Process," Mathematics, MDPI, vol. 9(19), pages 1-27, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:19:p:2470-:d:649256
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    References listed on IDEAS

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