A new integral equation for the first passage time density of the Ornstein-Uhlenbeck process

The Laplace transform of the first passage time density of the Ornstein--Uhlenbeck process for a constant threshold contains a ratio of two parabolic cylinder functions for which no analytical inversion formula is available. Recently derived inverse Laplace transforms for the product of two parabolic cylinder functions together with the convolution theorem of the Laplace transform then allow to derive a new Volterra integral equation for this first passage time density. The kernel of this integral equation contains a parabolic cylinder function and the Fortet renewal equation for the Ornstein-Uhlenbeck process emerges as a special case, namely when the order q of the parabolic cylinder function is set at 0. The integral equation is shown to hold both for constant as well as time dependent thresholds. Moreover, the kernel of the integral equation is regular for q<=-1.

• Publication year

  2019

• Venue

  arXiv: Probability

• Publication date

  2019-08-06

• Fields of study

  Mathematics, Physics

• Identifiers
  arXiv 1908.02071
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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