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

Part of: Markov processes Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  04 January 2016

Martin Forde ,
Andrey Pogudin  and
Hongzhong Zhang
Martin Forde*
Affiliation:
King's College London
Andrey Pogudin*
Affiliation:
King's College London
Hongzhong Zhang*
Affiliation:
Columbia University
*
∗ Postal address: Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK.
∗ Postal address: Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK.
∗∗∗∗ Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA. Email address: hzhang@stat.columbia.edu
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Abstract

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We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form d Xt=σ(Xt,Xt)dWt, where Xt= m ∧ inf0 ≤s≤tXs. 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 (Xt,Xt) 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.

Keywords

One-dimensional diffusionoccupation time formulastochastic functional differential equationdiffusion with memory

MSC classification

Primary: 60J60: Diffusion processes 47D07: Markov semigroups and applications to diffusion processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 860 - 875
Copyright
© Applied Probability Trust 

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