×

Invariant measures and boundedness in the mean for stochastic equations driven by Lévy noise. (English) Zbl 1502.60095

Summary: Existence of invariant measures and average stability in the mean are studied for stochastic differential equations driven by Lévy process. In particular, some natural conditions are found that verify stabilization of the equation (in the sense of the existence of invariant measures) by jump noise terms. These conditions are verified in several examples.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60J60 Diffusion processes
93E15 Stochastic stability in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albeverio, S., Di Persio, L., Mastrogiacomo, E. and Smii, B., A class of Lévy driven SDEs and their explicit invariant measures, Potential Anal.45 (2016) 229-259. · Zbl 1350.60049
[2] Applebaum, D., Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge Univ. Press, 2009). · Zbl 1200.60001
[3] Applebaum, D., Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes, Probab. Surveys12 (2015) 33-54. · Zbl 1323.60104
[4] Applebaum, D. and Siakalli, M., Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab.46 (2009) 1116-1129. · Zbl 1185.60058
[5] Applebaum, D. and Siakalli, M., Stochastic stabilization of dynamical systems using Lévy noise, Stoch. Dyn.10 (2010) 509-527. · Zbl 1218.93106
[6] Bhan, C., Chakraborty, P. and Mandrekar, V. S., Invariant measure and stability of the solution of a stochastic differential equation driven by a jump Lévy process, Int. J. Contemp. Math. Sci.7 (2012) 33-52. · Zbl 1266.60100
[7] Brockwell, J. P., Lévy-driven CARMA processes, Ann. Inst. Statist. Math.53 (2001) 113-124. · Zbl 0995.62089
[8] Grigoriu, M., Lyapunov exponents for nonlinear systems with Poisson white noise, Phys. Lett. A217 (1996) 258-262. · Zbl 0972.37520
[9] Jurek, Z. J., An integral representation of operator-self decomposable random variables, Bull. Acad. Pol. Sci.30 (1982) 385-393. · Zbl 0503.60063
[10] Kadlec, K. and Maslowski, B., Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process, Discrete Contin. Dyn. Syst.25 (2000) 4039-4055. · Zbl 1466.93179
[11] Kadlec, K. and Maslowski, B., Ergodic control for Lévy-driven linear stochastic equations in Hilbert spaces, Appl. Math. Optim.79 (2019) 547-565. · Zbl 1427.60127
[12] Khasminskii, R. Z., Stochastic Stability of Differential Equations (Sijthofff and Noordhoff, 1980). · Zbl 0441.60060
[13] Krylov, N. and Bogolyubov, N., La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire, Ann. Math.38 (1937) 65-113. · Zbl 0016.08604
[14] Kumar, U. and Riedle, M., Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process, J. Math. Anal. Appl.493 (2021) 124536. · Zbl 1466.60131
[15] Leha, G., Maslowski, B. and Ritter, G., Stability of solutions to semilinear stochastic evolution equations, Stochastic Anal. Appl.17 (1999) 1009-1051. · Zbl 0943.60049
[16] Li, C. W., Dong, Z. and Situ, R., Almost sure stability of linear stochastic differential equations with jumps, Probab. Theory Related Fields123 (2002) 121-155. · Zbl 1019.34055
[17] Mao, X., Stochastic Differential Equations and Applications, 2nd edn. (Elsevier, 2008).
[18] Mao, X. and Rodkina, A. E., Exponential stability of stochastic differential equations driven by discontinuous semimartingales, Stoch. Int. J. Probab. Stoch. Process.55 (1995) 207-224. · Zbl 0886.60058
[19] Nane, E. and Ni, Y., Stability of the solution of stochastic differential equation driven by time-changed Lévy noise, Proc. Amer. Math. Soc.145 (2017) 3085-3104. · Zbl 1364.65015
[20] Øksendal, B. K. and Sulem, A., Applied Stochastic Control of Jump Diffusions (Springer, 2007). · Zbl 1116.93004
[21] Patel, A. and Kosko, B., Stochastic resonance in continuous and spiking neuron models with Lévy noise, IEEE Trans. Neural Netw.19 (2008) 1993-2008.
[22] Tankov, P., Financial Modelling with Jump Processes (Chapman and Hall/CRC, 2003). · Zbl 1052.91043
[23] Zhang, X., Xu, Y., Schmalfuß, B., and Pei, B., Random attractors for stochastic differential equations driven by two-sided Lévy processes, Stochastic Anal. Appl.37 (2019) 1028-1041. · Zbl 1428.37050
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.