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 \(\mathbb{E}_x [e^{ - \theta T_d - \lambda } \int_0^{T_d } {1_{a < X_s < b} ds} ;T_d < T_c ]\), where T x 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.
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References
Borodin A N, Salminen P. Handbook of Brownian Motion—Facts and Formulae. 2nd ed. Basel: Birkhäuser Verlag, 2002
Buchholz H, Lichtblau H, Wetzel K. The Confluent Hypergeometric Function: with Special Emphasis on Its Applications. Berlin: Springer, 1969
Darling D A, Siegert A J F. The first passage problem for a continuous Markov process. Ann Math Statist, 1953, 24: 624–639
Feller W. Diffusion processes in one dimension. Trans Amer Math Soc, 1954, 77: 1–31
Gīhman Ĭ Ī, Skorohod A V. Stochastic Differential Equations. New York-Heidelberg: Springer-Verlag, 1972
Ito K, McKean H P. Diffusion Processes and Their Sample Paths. Berlin: Springer-Verlag, 1974
Landriault D, Renaud J-F, Zhou X. Occupation times of spectrally negative Lévy processes with applications. Stochastic Process Appl, 2011, 121: 2629–2641
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
Lejay A. On the constructions of the skew Brownian motion. Probab Surv, 2006, 3: 413–466
Li B, Zhou X. The joint Laplace transforms for diffusion occupation times. Adv Appl Probab, 2013, 45: 1–19
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
Pitman J, Yor M. Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches. Bernoulli, 2003, 9: 1–24
Prokhorov Yu V, Shiryaev A N. Probability Theory. III. Stochastic Calculus. Berlin: Springer-Verlag, 1998
Stroock D, Varadhan S R S. Diffusion processes with continuous coefficients. I. Comm Pure Appl Math, 1969, 22: 345–400
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Li, Y., Wang, S., Zhou, X. et al. Diffusion occupation time before exiting. Front. Math. China 9, 843–861 (2014). https://doi.org/10.1007/s11464-014-0402-6
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DOI: https://doi.org/10.1007/s11464-014-0402-6
Keywords
- Diffusion
- exit problem
- occupation time
- occupation density
- local time
- Brownian motion with two-valued drift
- skew Ornstein-Uhlenbeck (OU) process