On the first hitting time density for a reducible diffusion process

Abstract

In this paper, we study the classical problem of the first hitting time density to a moving boundary for a diffusion process, which satisfies the Cherkasov condition, and hence, can be reduced to a standard Wiener process. We give two complementary (forward and backward) formulations of this problem and provide semi-analytical solutions for both. By using the method of heat potentials, we show how to reduce these problems to linear Volterra integral equations of the second kind. For small values of t, we solve these equations analytically by using Abel equation approximation; for larger t we solve them numerically. We illustrate our method with representative examples, including Ornstein–Uhlenbeck processes with both constant and time-dependent coefficients. We provide a comparison with other known methods for finding the hitting density of interest, and argue that our method has considerable advantages and provides additional valuable insights. We also show applications of the problem and our method in various areas of financial mathematics.

Keywords:

• First hitting time density
• Cherkasov condition
• Method of heat potentials
• Volterra integral equation
• Abel integral equation
• Ornstein–Uhlenbeck process
• Pairs trading

Acknowledgements

We are grateful for helpful comments and constructive suggestions from anonymous referees.

Disclosure statement

No potential conflict of interest was reported by the authors.