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On the heat equation with a moving boundary and applications to hitting times for Brownian motion

Author

Listed:
  • Gerardo Hernández-del-Valle

    (CEMLA)

  • Wincy A. Guerra-Polania

    (Universidad EIA)

Abstract

In this paper we provide conditions under which the hitting-time problem for Brownian motion is equivalent to solving a heat equation with moving boundary and distributional initial conditions. Motivated by the hitting time problem, we study the heat equation with absorbing moving boundaries. Using Fourier analysis we develop a procedure to solve this problem for a family of curves that includes the square root, quadratic, and cubic boundaries. As an application of our results, and using Sturm-Liouvile theory, we compute the density of the hitting time of a Brownian motion to a family of quadratic boundaries.

Suggested Citation

  • Gerardo Hernández-del-Valle & Wincy A. Guerra-Polania, 2023. "On the heat equation with a moving boundary and applications to hitting times for Brownian motion," CEMLA Working Paper Series 05/2023, CEMLA.
    • Handle: RePEc:cml:wpseri:05
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    References listed on IDEAS

    as
    1. Nan Chen & S. G. Kou, 2009. "Credit Spreads, Optimal Capital Structure, And Implied Volatility With Endogenous Default And Jump Risk," Mathematical Finance, Wiley Blackwell, vol. 19(3), pages 343-378, July.
    2. Naoto Kunitomo & Masayuki Ikeda, 1992. "Pricing Options With Curved Boundaries1," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 275-298, October.
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