Abstract
In this paper, for homogeneous diffusion processes, the approach of Y. Li and X. Zhou [Statist. Probab. Lett., 2014, 94: 48–55] is adopted to find expressions of potential measures that are discounted by their joint occupation times over semi-infinite intervals (-∞, α) and (α, ∞): The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions for Brownian motion with drift, skew Brownian motion, and Brownian motion with two-valued drift, respectively.
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Acknowledgements
The authors would like to thank the anonymous referees for helpful suggestions and comments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11171044, 11571052, 11671132), the Natural Science Foundation of Hunan Province (Grant No. 2016JJ4061), the Scientific Research Project of Hunan University of Arts and Science (Grant No. 15ZD05), and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (2015)
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Chen, Y., Li, Y. & Zhou, X. An occupation time related potential measure for diffusion processes. Front. Math. China 12, 559–582 (2017). https://doi.org/10.1007/s11464-017-0625-4
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DOI: https://doi.org/10.1007/s11464-017-0625-4
Keywords
- Laplace transform
- occupation time
- potential measure
- exit time
- time-homogeneous diffusion
- Brownian motion with two-valued drift
- skew Brownian motion