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Kordzakhia NE, Novikov AA. On Maximal Inequalities for Ornstein--Uhlenbeck Processes with Jumps. THEORY OF PROBABILITY AND ITS APPLICATIONS 2022. [DOI: 10.1137/s0040585x97t990691] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time. J THEOR PROBAB 2020. [DOI: 10.1007/s10959-020-01040-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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On first passage times of sticky reflecting diffusion processes with double exponential jumps. J Appl Probab 2020. [DOI: 10.1017/jpr.2019.93] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
AbstractWe explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.
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Czarna I, Pérez JL, Rolski T, Yamazaki K. Fluctuation theory for level-dependent Lévy risk processes. Stoch Process Their Appl 2019. [DOI: 10.1016/j.spa.2019.03.006] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Zhou J, Wu L, Bai Y. Occupation times of Lévy-driven Ornstein–Uhlenbeck processes with two-sided exponential jumps and applications. Stat Probab Lett 2017. [DOI: 10.1016/j.spl.2017.01.021] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Abstract
In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.
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Xing X, Zhang W, Wang Y. The Stationary Distributions of Two Classes of Reflected Ornstein–Uhlenbeck Processes. J Appl Probab 2016. [DOI: 10.1239/jap/1253279847] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.
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BO LIJUN, REN GUIJUN, WANG YONGJIN, YANG XUEWEI. FIRST PASSAGE TIMES OF REFLECTED GENERALIZED ORNSTEIN–UHLENBECK PROCESSES. STOCH DYNAM 2012. [DOI: 10.1142/s0219493712500141] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
We study first passage problems of a class of reflected generalized Ornstein–Uhlenbeck processes without positive jumps. By establishing an extended Dynkin's formula associated with the process, we derive that the joint Laplace transform of the first passage time and an integral functional stopped at the time satisfies a truncated integro-differential equation. Two solvable examples are presented when the driven Lévy process is a drifted-Brownian motion and a spectrally negative stable process with index α ∈ (1, 2], respectively. Finally, we give two applications in finance.
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Affiliation(s)
- LIJUN BO
- Department of Mathematics, Xidian University, Xi'an 710071, China
| | - GUIJUN REN
- Global Markets Analytics, Opera Solutions, LLC, Shanghai 201204, China
| | - YONGJIN WANG
- School of Business, Nankai University, Tianjin 300071, China
| | - XUEWEI YANG
- School of Management and Engineering, Nanjing University, Nanjing 210093, China
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Novikov AA. On Distributions of First Passage Times and Optimal Stopping of AR(1) Sequences. THEORY OF PROBABILITY AND ITS APPLICATIONS 2009. [DOI: 10.1137/s0040585x97983730] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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Jacobsen M, Jensen AT. Exit times for a class of piecewise exponential Markov processes with two-sided jumps. Stoch Process Their Appl 2007. [DOI: 10.1016/j.spa.2007.01.005] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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