×

zbMATH — the first resource for mathematics

Asymptotic properties of jump-diffusion processes with state-dependent switching. (English) Zbl 1191.60091
The author deals with a class of jump-diffusion processes with state-dependent switching. A jump-diffusion process with state-dependent switching is a combination of a diffusion process with both random jumping and state-dependent switching, it is often called a hybrid jump-diffusion process or a jump-diffusion process with regime switching. First, the existence and uniqueness of the solution of a system of stochastic integro-differential equations are obtained with the aid of successive construction methods. Next, the non-explosiveness is proved by truncation arguments. Then, the Feller continuity is established by means of introducing some auxiliary processes and by making use of the Radon-Nikodym derivatives. Furthermore, the strong Feller continuity is proved by virtue of the relation between the transition probabilities of jump-diffusion processes and the corresponding diffusion processes. Finally, on the basis of the above results, the exponential ergodicity is obtained under the Foster-Lyapunov drift conditions. Two examples are also provided for illustration.

MSC:
60J60 Diffusion processes
60J27 Continuous-time Markov processes on discrete state spaces
34D23 Global stability of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Basak, G.K.; Bisi, A.; Ghosh, M.K., Stability of a random diffusion with linear drift, J. math. anal. appl., 202, 604-622, (1996) · Zbl 0856.93102
[2] Basak, G.K.; Bisi, A.; Ghosh, M.K., Stability of a degenerate diffusions with state-dependent switching, J. math. anal. appl., 240, 219-248, (1999) · Zbl 0939.93038
[3] Buffington, J.; Elliott, R.J., American options with regime switching, Internat. J. theoret. appl. finance, 5, 497-514, (2002) · Zbl 1107.91325
[4] Chen, M.F., From Markov chains to non-equilibrium particle systems, (1992), World Scientific Singapore
[5] Chen, Z.Q.; Zhao, Z., Switched diffusion processes and systems of elliptic equations—a Dirichlet space approach, Proc. roy. soc. edinb., 124A, 673-701, (1994) · Zbl 0807.47056
[6] Chen, Z.Q.; Zhao, Z., Potential theory for elliptic systems, Ann. probab., 24, 293-319, (1996) · Zbl 0854.60062
[7] Chen, Z.Q.; Zhao, Z., Harnack principle for weakly coupled elliptic systems, J. differential equations, 139, 261-282, (1997) · Zbl 0882.35039
[8] Chow, Y.S.; Teicher, H., Probability theory, (1978), Springer-Verlag New York
[9] Di Masi, G.B.; Kabanov, Y.M.; Runggaldier, W.J., Mean variance hedging of options on stocks with Markov volatility, Theory probab. appl., 39, 173-181, (1994) · Zbl 0836.60075
[10] Eizenberg, A.; Freidlin, M.I., On the Dirichlet problems for a class of second order PDE systems with small parameter, Stoch. stoch. rep., 33, 111-148, (1990) · Zbl 0723.60095
[11] Eizenberg, A.; Freidlin, M.I., Large deviations for Markov processes corresponding to PDE systems, Ann. probab., 21, 1015-1044, (1993) · Zbl 0776.60037
[12] Eizenberg, A.; Freidlin, M.I., Averaging principle for perturbed random evolution equations and corresponding Dirichlet problems, Probab. theory related fields, 94, 335-374, (1993) · Zbl 0767.60022
[13] Freidlin, M.I.; Wentzell, A.D., Random perturbations of dynamical systems, (1984), Springer-Verlag New York · Zbl 0522.60055
[14] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam · Zbl 0495.60005
[15] Khas’minskii, R.Z., Stochastic stability of differential equations, (1980), Stijhoff and Noordhoff Alphen · Zbl 0441.60060
[16] Khas’minskii, R.Z.; Zhu, C.; Yin, G., Stability of regime-switching diffusions, Stochastic process. appl., 117, 1037-1051, (2007) · Zbl 1119.60065
[17] Krishnamurthy, V.; Wang, X.; Yin, G., Spreading code optimization and adaptation in CDMA via discrete stochastic approximation, IEEE trans. automat. control, 50, 1927-1949, (2004) · Zbl 1297.94011
[18] Kunita, H., Supports of diffusion processes and controllability problems, (), 163-185
[19] Mao, X.; Yin, G.; Yuan, C., Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43, 264-273, (2007) · Zbl 1111.93082
[20] Meyn, S.P.; Tweedie, R.L., Stability of Markovian processes I: criteria for discrete time chains, Adv. appl. probab., 24, 542-574, (1992) · Zbl 0757.60061
[21] Meyn, S.P.; Tweedie, R.L., Stability of Markovian processes II: continuous time processes and sampled chains, Adv. appl. probab., 25, 487-517, (1993) · Zbl 0781.60052
[22] Meyn, S.P.; Tweedie, R.L., Stability of Markovian processes III: foster – lyapunov criteria for continuous-time processes, Adv. appl. probab., 25, 518-548, (1993) · Zbl 0781.60053
[23] Meyn, S.P.; Tweedie, R.L., Markov chains and stochastic stability, (1993), Springer-Verlag Berlin · Zbl 0925.60001
[24] Shiga, T.; Tanaka, H., Central limit theorem for a system of markovian particles with Mean field interactions, Z. wahr. verw. geb., 69, 439-459, (1985) · Zbl 0607.60095
[25] Skorohod, A.V., Asymptotic methods in the theory of stochastic differential equations, (1989), Amer. Math. Soc. Providence
[26] Wee, I.S., Stability for multidimensional jump-diffusion processes, Stochastic process. appl., 80, 193-209, (1999) · Zbl 0962.60046
[27] Wu, L.M., Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems, Stochastic process. appl., 91, 205-238, (2001) · Zbl 1047.60059
[28] Xi, F.B., Stability for a random evolution equation with Gaussian perturbation, J. math. anal. appl., 272, 458-472, (2002) · Zbl 1006.60056
[29] Xi, F.B., Stability of a random diffusion with nonlinear drift, Statist. probab. lett., 68, 273-286, (2004) · Zbl 1071.60070
[30] Xi, F.B., Invariant measure for the Markov process corresponding to a PDE system, Acta math. sin. (engl. ser.), 21, 457-464, (2005) · Zbl 1084.60018
[31] Xi, F.B., On the stability of a jump-diffusions with Markovian switching, J. math. anal. appl., 341, 588-600, (2008) · Zbl 1138.60044
[32] Xi, F.B., Feller property and exponential ergodicity of diffusion processes with state-dependent switching, Sci. China, ser. A, 51, 329-342, (2008) · Zbl 1141.60047
[33] Xi, F.B.; Zhao, L.Q., On the stability of diffusion processes with state-dependent switching, Sci. China, ser. A, 49, 1258-1274, (2006) · Zbl 1107.60321
[34] Yin, G.; Liu, R.H.; Zhang, Q., Recursive algorithms for stock liquidation: A stochastic optimization approach, SIAM J. optim., 13, 240-263, (2002) · Zbl 1021.91022
[35] Yin, G.; Song, Q.; Zhang, Z., Numerical solutions for jump-diffusions with regime switching, Stochastics, 77, 61-79, (2005) · Zbl 1071.60050
[36] Yuan, C.G.; Mao, X.R., Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic process appl., 103, 277-291, (2003) · Zbl 1075.60541
[37] Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM J. control optim., 46, 1155-1179, (2007) · Zbl 1140.93045
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.