Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T04:06:38.876Z Has data issue: false hasContentIssue false
  • English
  • Français

The spectral representation of Bessel processes with constant drift: applications in queueing and finance

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 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: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of order ν > −1 with constant drift μ ≠ 0. When ν > -½ and μ < 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalue λ 0 = 0, followed by an infinite series of terms corresponding to the higher eigenvalues λ n , n = 1,2,…, as well as an integral over the continuous spectrum above μ 2/2. When −1 < ν < -½ and μ < 0, there is only one eigenvalue λ 0 = 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

Keywords

Bessel processpole-seeking Brownian motionCoulomb potentialspectral expansionheavy traffic limitCIR modelmodelinterest-rate model

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
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 327 - 344
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.CrossRefGoogle Scholar
Abramowitz, M., and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Ahn, D.-H., and Gao, B. (1999). A parametric nonlinear model of term structure dynamics. Rev. Financial Studies 12, 721–762.CrossRefGoogle Scholar
Alili, L., Patie, P., and Pedersen, J. L. (2003). Hitting time of a fixed level by an OU process. Working Paper, ETH Zurich.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. Springer, Berlin.CrossRefGoogle Scholar
Chan, K. C., Karolyi, G. A., Longstaff, F. A., and Sanders, A. B. (1992). An empirical comparison of alternative models of the short rate. J. Finance 47, 1209–1228.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.CrossRefGoogle Scholar
Davydov, D., and Linetsky, V. (2003). Pricing options on scalar diffusions: an eigenfunction expansion approach. Operat. Res. 51, 185–209.CrossRefGoogle Scholar
Delbaen, F., and Yor, M. (2002). Passport options. Math. Finance 12, 299–328.CrossRefGoogle Scholar
Duffie, D., and Singleton, K. (1999). Modeling term structures of defaultable bonds. Rev. Financial Studies 12, 687–720.CrossRefGoogle Scholar
Feller, W. (1951). Two singular diffusion problems. Ann. Math. 54, 173–82.CrossRefGoogle Scholar
Göing-Jaeschke, A., and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9, 313–349.CrossRefGoogle 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.CrossRefGoogle Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
Heston, S. L. (1997). A simple new formula for options with stochastic volatility. Working Paper, Washington University.Google Scholar
Itô, K. and McKean, H. P. (1974). Diffusion Processes and Their Sample Paths, 2nd edn. Springer, Berlin.Google Scholar
Kendall, D.G. (1974). Pole-seeking Brownian motion and bird navigation. J. R. Statist. Soc. B 36, 365–417.Google Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.CrossRefGoogle Scholar
Kou, S. C., and Kou, S. G. (2002). A diffusion model for growth stocks. To appear in Math. Operat. Res.Google Scholar
Kyprianou, A. E., and Pistorius, M. R. (2003). Perpetual options and Canadization through fluctuation theory. Ann. Appl. Prob. 13, 1077–1098.CrossRefGoogle Scholar
Langer, H., and Schenk, W. S. (1990). Generalized second-order differential operators, corresponding gap diffusions and superharmonic transformations. Math. Nachr. 148, 7–45.CrossRefGoogle Scholar
Levitan, B. M., and Sargsjan, I. S. (1975). Introduction to Spectral Theory. American Mathematical Society, Providence, RI.Google Scholar
Lewis, A. (1994). Three expansion regimes for interest rate term structure models. Working paper.Google Scholar
Lewis, A. (1998). Applications of eigenfunction expansions in continuous-time finance. Math. Finance 8, 349–383.CrossRefGoogle Scholar
Lewis, A. (2000). Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach, CA.Google Scholar
Linetsky, V. (2001). Spectral expansions for Asian (average price) options. To appear in Operat. Res. Available at http://users.iems.nwu.edu/∼linetsky/.Google Scholar
Linetsky, V. (2002a). Lookback options and diffusion hitting times: a spectral expansion approach. To appear in Finance Stoch. Available at http://users.iems.nwu.edu/∼linetsky/.Google Scholar
Linetsky, V. (2002b). The spectral decomposition of the option value. Working Paper, Northwestern University. Available at http://users.iems.nwu.edu/∼linetsky/.Google Scholar
Linetsky, V. (2003). Computing hitting time densities for OU and CIR processes: applications to mean-reverting models. To appear in J. Comput. Finance. Available at http://users.lems.nwu.edu/∼linetsky/.Google Scholar
Matsumoto, H., and Yor, M. (2000). An analogue of Pitman's 2M-X theorem for exponential Wiener functionals. I. A time-inversion approach. Nagoya Math. J. 159, 125–166.CrossRefGoogle Scholar
McKean, H. (1956). Elementary solutions for certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82, 519–548.CrossRefGoogle Scholar
Morse, P. M., and Feshbach, H. (1953). Methods of Theoretical Physics, Vol. 2. McGraw-Hill, New York.Google Scholar
Pitman, J.W., and Yor, M. (1981). Bessel processes and infinitely divisible laws. In Stochastic Integrals (Lecture Notes Math. 851), ed. Williams, D., Springer, Berlin, pp. 285–370.CrossRefGoogle Scholar
Pitman, J.W., and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrscheinlichkeitsth. 59, 425–457.CrossRefGoogle Scholar
Revuz, D., and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn, Springer, Berlin.CrossRefGoogle Scholar
Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.Google Scholar
Titchmarsh, E. C. (1962). Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon, Oxford.CrossRefGoogle Scholar
Watanabe, S. (1975). On time inversion of one-dimensional diffusion processes. Z. Wahrscheinlichkeitsth. 31, 115–124.CrossRefGoogle Scholar
Yor, M. (1984). On square-root boundaries for Bessel processes, and pole-seeking Brownian motion. Stochastic Analysis and Applications (Lecture Notes Math. 1095), eds Truman, A. and Williams, D., Springer, Berlin, pp. 100–107.CrossRefGoogle Scholar