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On the transition densities for reflected diffusions

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Vadim Linetsky
Vadim Linetsky*
Affiliation:
Northwestern University
*
∗ Postal address: Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA. Email address: linetsky@iems.northwestern.edu
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Abstract

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Diffusion models in economics, finance, queueing, mathematical biology, and electrical engineering often involve reflecting barriers. In this paper, we study the analytical representation of transition densities for reflected one-dimensional diffusions in terms of their associated Sturm-Liouville spectral expansions. In particular, we provide explicit analytical expressions for transition densities of Brownian motion with drift, the Ornstein-Uhlenbeck process, and affine (square-root) diffusion with one or two reflecting barriers. The results are easily implementable on a personal computer and should prove useful in applications.

Keywords

Reflected diffusionreflected Brownian motionreflected Ornstein-Uhlenbeck processreflected affine processspectral expansioncurrency target zoneheavy traffic

MSC classification

Primary: 60J35: Transition functions, generators and resolvents 60J60: Diffusion processes 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 435 - 460
Copyright
Copyright © Applied Probability Trust 2005 

References

Abate, J. and Whitt, W. (1987a). Transient behavior of regulated Brownian motion. I. Starting at the origin. Adv. Appl. Prob. 19, 560–598.CrossRefGoogle Scholar
Abate, J. and Whitt, W. (1987b). Transient behavior of regulated Brownian motion. II. Nonzero initial conditions. Adv. Appl. Prob. 19, 599–631.Google Scholar
Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Ball, C. A. and Roma, A. (1998). Detecting mean reversion within reflecting barriers: applications to the European exchange rate mechanism. Appl. Math. Finance 5, 1–15.Google Scholar
Bertolla, G. and Caballero, R. J. (1992). Target zones and realignments. Amer. Econom. Rev. 82, 520–536.Google Scholar
Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion. Birkhäuser, Boston, MA.Google Scholar
Buchholz, H. (1969). The Confluent Hypergeometric Function with Special Emphasis on Its Applications (Springer Tracts Natural Philos. 15). Springer, New York.Google Scholar
Coffman, E. G., Puhalskii, A. A. and Reiman, M. I. (1998). Polling systems in heavy traffic: a Bessel process limit. Math. Operat. Res. 23, 257–304.CrossRefGoogle Scholar
Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.Google Scholar
Davydov, D. and Linetsky, V. (2003). Pricing options on scalar diffusions: an eigenfunction expansion approach. Operat. Res. 51, 185–209.Google Scholar
De Jong, F. (1994). A univariate analysis of European monetary system exchange rates using a target zone model. J. Appl. Econometrics 9, 31–45.Google Scholar
Erdelyi, A. (1953). Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York.Google Scholar
Farnsworth, H. and Bass, R. (2003). The term structure with semi-credible targeting. J. Finance 58, 839–865.Google Scholar
Feller, W. (1951). Two singular diffusion problems. Ann. Math. 54, 173–182.Google Scholar
Fulton, C. and Pruess, S. (1994). Eigenvalue and eigenfunction asymptotics for regular Sturm–Liouville problems. J. Math. Anal. Appl. 188, 297–340.Google Scholar
Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9, 313–349.Google Scholar
Goldstein, R. and Keirstead, W. P. (1997). On the term structure of interest rates in the presence of reflecting and absorbing boundaries. Res. Rep. 97-1, Ohio State University.Google Scholar
Gorovoi, V. and Linetsky, V. (2004). Black's model of interest rates as options, eigenfunction expansions and Japanese interest rates. Math. Finance 14, 49–78.Google Scholar
Hanson, S. D., Myers, R. J. and Hilker, J. H. (1999). Hedging with futures and options under a truncated cash price distribution. J. Agricultural Appl. Econom. 31, 449–459.Google Scholar
Harrison, M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
Itô, K. and McKean, H. (1974). Diffusion Processes and their Sample Paths. Springer, Berlin.Google Scholar
Karlin, S. and McGregor, J. (1957). The differential equations of birth-and-death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, San Diego, CA.Google Scholar
Krugman, P. R. (1991). Target zones and exchange rate dynamics. Quart. J. Econom. 106, 669–682.Google Scholar
Langer, H. and Schenk, W. S. (1990). Generalized second-order differential operators, corresponding gap diffusions and superharmonic transformations. Math. Nachr. 148, 7–45.Google Scholar
Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.Google Scholar
Levinson, N. (1951). A simplified proof of the expansion theorem for singular second order differential operators. Duke Math. J. 18, 57–71.Google Scholar
Levitan, B. M. (1950). Expansion in Characteristic Functions of Differential Equations of the Second Order. Gostekhizdat, Moscow (in Russian).Google Scholar
Levitan, B. M. and Sargsjan, I. S. (1975). Introduction to Spectral Theory. American Mathematical Society, Providence, RI.Google Scholar
Lewis, A. (1998). Applications of eigenfunction expansions in continuous-time finance. Math. Finance 8, 349–383.Google Scholar
Linetsky, V. (2004a). Computing hitting times of OU and affine diffusions: applications to mean-reverting models. J. Comput. Finance 7, 1–22.Google Scholar
Linetsky, V. (2004b). Lookback options and diffusion hitting times: a spectral expansion approach. Finance Stoch. 8, 373–398.Google Scholar
Linetsky, V. (2004c). Spectral expansions for Asian (average price) options. Operat. Res. 52, 856–867.Google Scholar
Linetsky, V. (2004d). The spectral decomposition of the option value. Internat. J. Theoret. Appl. Finance 7, 337–384.Google Scholar
Linetsky, V. (2004e). The spectral representation of Bessel processes with drift: applications in queueing and finance. J. Appl. Prob. 41, 327–344.CrossRefGoogle Scholar
McKean, H. (1956). Elementary solutions for certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82, 519–548.CrossRefGoogle Scholar
Pitman, J.W. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrscheinlichkeitsth. 59, 425–457.Google Scholar
Ricciardi, L. M. and Sacerdote, L. (1987). On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary. J. Appl. Prob. 24, 355–369.Google Scholar
Schobel, R. and Zhu, J. (1999). Stochastic volatility with an Ornstein–Uhlenbeck process: an extension. Europ. Finance Rev. 3, 23–46.Google Scholar
Shiga, T. and Watanabe, S. (1973). Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrscheinlichkeitsth. 27, 37–46.Google Scholar
Srikant, R. and Whitt, W. (1996). Simulation run lengths to estimate blocking probabilities. ACM Trans. Model. Comput. Simul. 6, 7–52.Google Scholar
Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.Google Scholar
Svensson, L. E. O. (1991). The term structure of interest rate differentials in a target zone. Theory and Swedish data. J. Monetary Econom. 28, 87–116.Google Scholar
Veestraeten, D. (2004). The conditional probability density function for a reflected Brownian motion. Comput. Econom. 23, 185–207.Google Scholar
Ward, A. R. and Glynn, P. W. (2003a). A diffusion approximation for a Markovian queue with reneging. Queueing Systems 43, 103–128.Google Scholar
Ward, A. R. and Glynn, P. W. (2003b). Properties of the reflected Ornstein–Uhlenbeck process. Queueing Systems 44, 109–123.Google Scholar
Whitt, W. (2004). A diffusion approximation for the G/GI/n/m queue. Operat. Res. 52, 922–941.CrossRefGoogle Scholar
Whitt, W. (2005). Heavy-traffic limits for the G/H2*/n/m queue. Math. Operat. Res. 30, 1–27.Google Scholar
Wong, E. (1964). The construction of a class of stationary Markoff processes. In Proc. Symp. Appl. Math. Vol. XVI, ed. Bellman, R., American Mathematical Society, Providence, RI, pp. 264–276.Google Scholar