Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-27T18:37:35.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition probabilities for some ‘special' diffusions

Published online by Cambridge University Press:  14 July 2016

Václav E. Beneš  and
Ioannis Karatzas
Václav E. Beneš*
Affiliation:
AT&T Bell Laboratories
Ioannis Karatzas*
Affiliation:
Columbia University
*
∗Postal address: AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
∗∗Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the transition probabilities of the diffusions dXt = (1 – exp(Xt))dt + dWt and dXt = – tanh Xtdt + dWt, in terms of special functions of mathematical physics (confluent hypergeometric and Legendre functions, respectively).

Keywords

GIRSANOV THEOREMFEYNMAN–KAC FORMULASPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 4 , December 1987 , pp. 888 - 898
Copyright
Copyright © Applied Probability Trust 1987 

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.)

Footnotes

Research supported by the Air Force Office of Scientific Research under grant AFOSR-86-0203.

References

Friedman, A. (1975) Stochastic Differential Equations and Applications , Vol. 1. Academic Press, New York.Google Scholar
Girsanov, I. V. (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Prob. Appl. 5, 285–301.CrossRefGoogle Scholar
Kac, M. (1951) On some connections between probability theory and differential and integral equations. Proc. 2nd Berkeley Symp. Math. Statist. Prob. , 189–215.Google Scholar
Kallianpur, G. (1980) Stochastic Filtering Theory. Springer-Verlag, New York.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1984) Trivariate density of Brownian motion, its local and occupation time, with application to stochastic control. Ann. Prob. 12, 819–828.Google Scholar
Landau, L. D. and Lifshitz, E. M. (1965) Quantum Mechanics: Non-relativistic Theory, 2nd edn. Pergamon Press, Oxford.Google Scholar
Magnus, W., Oberhettinger, F. and Soni, R. P. (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, New York.Google Scholar
Millman, J. and Halkias, C. C. (1972) Integrated Electronics: Analogue and Digital Circuits and Systems. McGraw Hill-Kogakusha, Tokyo.Google Scholar
Rosenblatt, M. (1951) On a class of Markov processes. Trans. Amer. Math. Soc. 71, 120–135.CrossRefGoogle Scholar
Varadhan, S. R. S. (1980) Lectures on Diffusion Problems and Partial Differential Equations. Lecture Notes, Tata Institute of Fundamental Research, Bombay. Springer-Verlag, Berlin.Google Scholar