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A cumulant approach for the first-passage-time problem of the Feller square-root process

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  • Di Nardo, Elvira
  • D’Onofrio, Giuseppe

Abstract

The paper focuses on an approximation of the first passage time probability density function of a Feller stochastic process by using cumulants and a Laguerre-Gamma polynomial approximation. The feasibility of the method relies on closed form formulae for cumulants and moments recovered from the Laplace transform of the probability density function and using the algebra of formal power series. To improve the approximation, sufficient conditions on cumulants are stated. The resulting procedure is made easier to implement by the symbolic calculus and a rational choice of the polynomial degree depending on skewness, kurtosis and hyperskewness. Some case studies coming from neuronal and financial fields show the goodness of the approximation even for a low number of terms. Open problems are addressed at the end of the paper.

Suggested Citation

  • Di Nardo, Elvira & D’Onofrio, Giuseppe, 2021. "A cumulant approach for the first-passage-time problem of the Feller square-root process," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    • Handle: RePEc:eee:apmaco:v:391:y:2021:i:c:s0096300320306603
    DOI: 10.1016/j.amc.2020.125707
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    References listed on IDEAS

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    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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    Cited by:

    1. Elvira Di Nardo & Giuseppe D’Onofrio, 2021. "On the Cumulants of the First Passage Time of the Inhomogeneous Geometric Brownian Motion," Mathematics, MDPI, vol. 9(9), pages 1-17, April.
    2. Meier, Christian & Li, Lingfei & Zhang, Gongqiu, 2023. "Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation," European Journal of Operational Research, Elsevier, vol. 305(3), pages 1292-1308.
    3. 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.
    4. Virginia Giorno & Amelia G. Nobile, 2021. "Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics," Mathematics, MDPI, vol. 9(16), pages 1-29, August.

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