RESEARCH ARTICLE

Diffusion occupation time before exiting

  • Yingqiu LI 1 ,
  • Suxin WANG 2 ,
  • Xiaowen ZHOU 3 ,
  • Na ZHU , 1
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  • 1. College of Mathematics and Computing Sciences, Changsha University of Science and Technology, Changsha 410004, China
  • 2. College of Mathematics, Nankai University, Tianjin 300071, China
  • 3. Department of Mathematics and Statistics, Concordia University, Montreal, Quebec H3G 1M8, Canada

Received date: 19 Feb 2014

Accepted date: 17 Jun 2014

Published date: 20 Aug 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

Using the approach of D. Landriault et al. and B. Li and X. Zhou, for a one-dimensional time-homogeneous diffusion process X and constants c<a<b<d, we find expressions of double Laplace transforms of the form Ex[e-θTd-λ0Td1a<Xs<bds;Td<Tc], where Tx denotes the first passage time of level x. As applications, we find explicit Laplace transforms of the corresponding occupation time and occupation density for the Brownian motion with two-valued drift and that of occupation time for the skew Ornstein-Uhlenbeck process, respectively. Some known results are also recovered.

Cite this article

Yingqiu LI , Suxin WANG , Xiaowen ZHOU , Na ZHU . Diffusion occupation time before exiting[J]. Frontiers of Mathematics in China, 2014 , 9(4) : 843 -861 . DOI: 10.1007/s11464-014-0402-6

1
Borodin A N, Salminen P. Handbook of Brownian Motion-Facts and Formulae. 2nd ed. Basel: Birkhäuser Verlag, 2002

DOI

2
Buchholz H, Lichtblau H, Wetzel K. The Confluent Hypergeometric Function: with Special Emphasis on Its Applications. Berlin: Springer, 1969

DOI

3
Darling D A, Siegert A J F. The first passage problem for a continuous Markov process. Ann Math Statist, 1953, 24: 624-639

DOI

4
Feller W. Diffusion processes in one dimension. Trans Amer Math Soc, 1954, 77: 1-31

DOI

5
Gīhman Ĭ Ī, Skorohod A V. Stochastic Differential Equations. New York-Heidelberg: Springer-Verlag, 1972

DOI

6
Ito K, McKean H P. Diffusion Processes and Their Sample Paths. Berlin: Springer-Verlag, 1974

7
Landriault D, Renaud J-F, Zhou X. Occupation times of spectrally negative Lévy processes with applications. Stochastic Process Appl, 2011, 121: 2629-2641

DOI

8
Le Gall J F. One-dimensional stochastic differential equations involving the local times of the unknown process. In: Truman A, Williams D, eds. Stochastic Analysis and Applications: Proceedings of the International Conference held in Swansea, April 11-15, 1983. Lecture Notes Math, Vol 1095. Berlin: Springer, 1984, 51-82

9
Lejay A. On the constructions of the skew Brownian motion. Probab Surv, 2006, 3: 413-466

DOI

10
Li B, Zhou X. The joint Laplace transforms for diffusion occupation times. Adv Appl Probab, 2013, 45: 1-19

DOI

11
Loeffen R L, Renaud J-F, Zhou Z. Occupation times of intervals until first passage times for spectrally negative Lévy processes. Stochastic Process Appl, 2014, 124: 1408-1435

DOI

12
Pitman J, Yor M. Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches. Bernoulli, 2003, 9: 1-24

DOI

13
Prokhorov Yu V, Shiryaev A N. Probability Theory. III. Stochastic Calculus. Berlin: Springer-Verlag, 1998

DOI

14
Stroock D, Varadhan S R S. Diffusion processes with continuous coefficients. I. Comm Pure Appl Math, 1969, 22: 345-400

DOI

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