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

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

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 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.

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

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 860 - 875
Copyright
© Applied Probability Trust 

References

Bertoin, J., Chaumont, L. and Pitman, J. (2003). Path transformations of first passage bridges. Electron. Commun. Prob. 8, 155–166.CrossRefGoogle Scholar
Bertoin, J., Pitman, J., and Ruiz de Chavez, J. (1999). Constructions of a Brownian path with a given minimum. Electron. Commun. Prob. 4, 31–37.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Brunick, G. and Shreve, S. (2012). Matching an Itô process by a solution of a stochastic differential equation. Submitted.Google Scholar
Carr, P. (2009). Local variance gamma option pricing model. Presentation, Bloomberg/Courant Institute.Google Scholar
Cox, A. M. G., Hobson, D. and Obłój, J. (2011). Time-homogeneous diffusions with a given marginal at a random time. ESAIM Prob. Statist. 15, S11–S24.CrossRefGoogle Scholar
Dermann, E., Ergener, D. and Kani, I. (1995). Static options replication. J. Derivatives 2, 78–95.CrossRefGoogle Scholar
Durrett, R. T., Iglehart, D. L. and Miller, D. R. (1977). Weak convergence to Brownian meander and Brownian excursion. Ann. Prob. 5, 117–129.Google Scholar
Figalli, A. (2008). Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254, 109–153.CrossRefGoogle Scholar
Forde, M. (2011). A diffusion-type process with a given Joint law for the terminal level and supremum at an independent exponential time. Stoch. Process. Appl. 121, 2802–2817.CrossRefGoogle Scholar
Imhof, J.-P. (1984). Density factorizations for Brownian motion, meander and the three dimensional Bessel process, and applications. J. Appl. Prob. 21, 500–510.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Mao, X. (1997). Stochastic Differential Equations and Their Applications. Horwood Publishing, Chichester.Google Scholar
Mohammed, S. E. A. (1984). Stochastic Functional Differential Equations. Pitman, Boston, MA.Google Scholar
Pauwels, E. J. (1987). Smooth first-passage densities for one-dimensional diffusions. J. Appl. Prob. 24, 370–377.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Rogers, L. C. G. (1985). Smooth transition densities for one-dimensional diffusions. Bull. London Math. Soc. 17, 157–161.CrossRefGoogle Scholar
Rogers, L. C. G. (1993). The Joint law of the maximum and the terminal value of a martingale. Prob. Theory Relat. Fields 95, 451–466.CrossRefGoogle Scholar
Rogers, L. C. G. (2012). Extremal martingales. Talk at EPSRC Symposium Workshop - Optimal Stopping, Optimal Control and Finance.Google Scholar
Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes and Martingales, Vol. 2. John Wiley, New York.Google Scholar
Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, Berlin.Google Scholar
Williams, D. (1974). Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. 28, 738–768.CrossRefGoogle Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press.CrossRefGoogle Scholar