Hitting Times, Occupation Times, Trivariate Laws and the Forward Kolmogorov Equation for a One-Dimensional Diffusion with Memory

• Published in: Advances in applied probability Vol. 45; no. 3; pp. 860 - 875
• Main Authors: Forde, Martin, Pogudin, Andrey, Zhang, Hongzhong
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01-09-2013; Applied Probability Trust
• Subjects: 47D07; 60J60; Boundaries; Brownian motion; Call options; Classification; Density; Differential equations; Diffusion; diffusion with memory; Equivalence; General Applied Probability; Laws of Motion; Martingales; Mathematical analysis; Mathematical functions; Mathematical theorems; Mathematics; Meanders; Occupation; occupation time formula; One-dimensional diffusion; Perceptron convergence procedure; Probability; Put & call options; stochastic functional differential equation; Stochastic models; Studies; Uniqueness; occupation time formula; One-dimensional diffusion; stochastic functional differential equation; diffusion with memory; 60J60; 47D07
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1239/aap/1377868542

We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form d X
t
=σ(X
t
,
X
t
)dW
t
, where
X
t
= m ∧ inf0 ≤s≤t
X
s
. In particular, we compute the expected time for X to leave an interval, classify the boundary behavior at 0, and derive a new occupation time formula for X. We also show that (X
t
,
X
t
) admits a joint density, which can be characterized in terms of two independent tied-down Brownian meanders (or, equivalently, two independent Bessel-3 bridges). Finally, we show that the joint density satisfies a generalized forward Kolmogorov equation in a weak sense, and we derive a new forward equation for down-and-out call options.