zbMATH — the first resource for mathematics

Jump-diffusions with state-dependent switching: existence and uniqueness, Feller property, linearization, and uniform ergodicity. (English) Zbl 1274.60253
The authors analyze a class of jump-diffusions with state-space dependent switching. In contrast to previous work their model allows for a characteristic measure which is \(\sigma\)-finite. ‘Existence and uniqueness of the underlying process are obtained by representing the switching component as a stochastic integral with respect to a Poisson random measure and by using a successive approximation method’. Furthermore the authors prove the Feller property. To this end they introduce auxiliary processes and they make use of Radon-Nikodym derivatives. Irreducibility and the fact that ‘all compact sets being petite are demonstrated. Based on these results, the uniform ergodicity is established under a general Lyapunov condition. Finally, easily verifiable conditions for uniform ergodicity are established when the jump-diffusions are linearizable with respect to the variable \(x\) (the state variable corresponding to the jump-diffusion component) in a neighborhood of the infinity’. The results are illustrated by some examples.

60J75 Jump processes (MSC2010)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J27 Continuous-time Markov processes on discrete state spaces
Full Text: DOI
[1] Basak G K, Bisi A, Ghosh M K. Stability of a degenerate diffusions with state-dependent switching. J Math Anal Appl, 1999, 240: 219–248 · Zbl 0939.93038 · doi:10.1006/jmaa.1999.6610
[2] Chen M F. From Markov Chains to Non-Equilibrium Particle Systems, 2nd ed. Singapore: World Scientific, 2004 · Zbl 1078.60003
[3] Down D, Meyn S P, Tweedie R L. Exponential and uniform ergodicity of Markov processes. Ann Probab, 1995, 23: 1671–1691 · Zbl 0852.60075 · doi:10.1214/aop/1176987798
[4] Ghosh M K, Arapostathis A, Marcus S I. Ergodic control of switching diffusions. SIAM J Contr Optim, 1997, 35: 1952–1988 · Zbl 0891.93081 · doi:10.1137/S0363012996299302
[5] Guyon X, Iovleff S, Yao J F. Linear diffusion with stationary switching regime. ESAIM Probab Stat, 2004, 8: 25–35 · Zbl 1033.60084 · doi:10.1051/ps:2003017
[6] Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes. Amsterdam: North-Holland, 1981 · Zbl 0495.60005
[7] Khasminskii R Z, Zhu C, Yin G. Stability of regime-switching diffusions. Stoch Process Appl, 2007, 117: 1037–1051 · Zbl 1119.60065 · doi:10.1016/j.spa.2006.12.001
[8] Kunita H. Supports of diffusion processes and controllability problems. In: Itô K, ed. Proc Internat Symp: Stochastic Differential Equations. New York: Wiley, 1978, 163–185 · Zbl 0409.60063
[9] Li C W, Dong Z, Situ R. Almost sure stability of linear stochastic differential equations with jumps. Probab Theory Relat Fields, 2002, 123: 121–155 · Zbl 1019.34055 · doi:10.1007/s004400200198
[10] Mao X. Stability of stochastic differential equations with Markovian switching. Stoch Process Appl, 1999, 79: 45–67 · Zbl 0962.60043 · doi:10.1016/S0304-4149(98)00070-2
[11] Meyn S P, Tweedie R L. Stability of Markovian processes I: Discrete time chains. Adv Appl Probab, 1992, 24: 542–574 · Zbl 0757.60061 · doi:10.2307/1427479
[12] Meyn S P, Tweedie R L. Stability of Markovian processes II: Continuous time processes and sampled chains. Adv Appl Probab, 1993, 25: 487–517 · Zbl 0781.60052 · doi:10.2307/1427521
[13] Meyn S P, Tweedie R L. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv Appl Probab, 1993, 25: 518–548 · Zbl 0781.60053 · doi:10.2307/1427522
[14] Meyn S P, Tweedie R L. Markov Chains and Stochastic Stability. Berlin: Springer-Verlag, 1993
[15] Musuda M. Ergodicity and exponential \(\beta\)-mixing bound for multidimensional diffusions with jumps. Stoch Process Appl, 2007, 117: 36–56
[16] Skorohod A V. Asymptotic Methods in the Theory of Stochastic Differential Equations. Providence, RI: Amer Math Soc, 1989
[17] Wang J. Criteria for ergodicity of Lévy type operators in dimension one. Stoch Process Appl, 2008, 118: 1909–1928 · Zbl 1157.60072 · doi:10.1016/j.spa.2007.11.003
[18] Wang J. Regularity of semigroups generated by Lévy type operators via coupling. Stoch Process Appl, 2010, 120: 1680–1700 · Zbl 1204.60071 · doi:10.1016/j.spa.2010.04.007
[19] Wee I S. Stability for multidimensional jump-diffusion processes. Stoch Process Appl, 1999, 80: 193–209 · Zbl 0962.60046 · doi:10.1016/S0304-4149(98)00078-7
[20] Xi F B. Asymptotic properties of jump-diffusion processes with state-dependent switching. Stoch Process Appl, 2009, 119: 2198–2221 · Zbl 1191.60091 · doi:10.1016/j.spa.2008.11.001
[21] Xi F B, Yin G. Asymptotic properties of a mean-field model with a continuous-state-dependent switching process. J Appl Probab, 2009, 46: 221–243 · Zbl 1159.60341 · doi:10.1239/jap/1238592126
[22] Xi F B, Zhao L Q. On the stability of diffusion processes with state-dependent switching. Sci China Ser A, 2006, 49: 1258–1274 · Zbl 1107.60321 · doi:10.1007/s11425-006-2017-1
[23] Yin G, Xi F B. Stability of regime-switching jump diffusions. SIAM J Contr Optim, 2010, 48: 4525–4549 · Zbl 1210.60089 · doi:10.1137/080738301
[24] Yin G, Zhu C. Hybrid Switching Diffusions: Properties and Applications. New York: Springer, 2010 · Zbl 1279.60007
[25] Yuan C G, Mao X R. Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch Process Appl, 2003, 103: 277–291 · Zbl 1075.60541 · doi:10.1016/S0304-4149(02)00230-2
[26] Zhu C, Yin G. Asymtotic properties of hybrid diffusion systems. SIAM J Contr Optim, 2007, 46: 1155–1179 · Zbl 1140.93045 · doi:10.1137/060649343
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.