Abstract
This work is devoted to the weak convergence analysis of a class of aggregated processes resulting from singularly perturbed switching diffusions with fast and slow motions. The processes consist of diffusion components and pure jump components. The states of the pure jump component are naturally divisible into a number of classes. Aggregate the states in each weakly irreducible class by a single state leading to an aggregated process. Under suitable conditions, it is shown that the aggregated process converges weakly to a switching diffusion process whose generator is an average with respect to the quasi-stationary distribution of the jump process.
Similar content being viewed by others
REFERENCES
Eidelman, S. D. (1969). Parabolic Systems, North-Holland, New York.
Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes: Characterization andConvergence, Wiley, New York.
Gihman, I. I., and Skorohod, A. V. (1979). Theory of Stochastic Processes, III, Springer-Verlag, Berlin.
Il'in, A. M., Khasminskii, R. Z., and G. Yin (1999). Singularly perturbed switching diffusions: Rapid switchings and fast diffusions. J. Optim. Theory Appl. 102, 555–591.
Khasminskii, R. Z., and Yin, G. (1996). Asymptotic series for singularly perturbed KolmogorovûFokkerûPlanck equations. SIAM J. Appl. Math. 56, 1766–1793.
Khasminskii, R. Z., Yin, G., and Zhang, Q. (1996). Asymptotic expansions of singularly perturbed systems involving rapidly fluctuating Markov chains. SIAM J. Appl. Math. 56, 277–293.
Khasminskii, R. Z., Yin, G., and Zhang, Q. (1997). Constructing asymptotic series for probability distribution of Markov chains with weak and strong interactions. Quart. Appl. Math. LV, 177-200.
Kushner, H. J. (1984). Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, Massachusetts.
Ladyzhenskaia, O. A., Solonnikov, V. A., and Ural'tseva, N. N. (1968). Linear andquasilinear equations of parabolic type, Trans. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, Rhode Island.
Liptser, R. S., and Shiryayev, A. N. (1997). Statistics of Random Processes I, Springer-Verlag, New York.
Pan, Z. G., and Ba?ar, T. (1995). H ?-control of Markovian jump linear systems and solutions to associated piecewise-deterministic differential games. In G. J. Olsder (Ed.), New Trends in Dynamic Games and Applications, Birkhäuser, Boston, pp. 61–94.
Phillips, R. G., and Kokotovic, P. V. (1981). A singular perturbation approach to modelling and control of Markov chains. IEEE Trans. Automat. Control 26, 1087–1094.
Sethi, S. P., and Zhang, Q. (1994). Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkhäuser, Boston.
Skorohod, A. V. (1989). Asymptotic Methods of the Theory of Stochastic Differential Equations, Trans. Math. Monographs, Vol. 78, Amer. Math. Soc., Providence.
Yin, G., and Kniazeva, M. (1999). Singularly perturbed multidimensional switching diffusions with fast and slow switches, J. Math. Anal. Appl. 229, 605–630.
Yin, G., and Zhang, Q. (1998). Continuous-time Markov Chains andApplications: A Singular Perturbations Approach, Springer-Verlag, New York.
Yin, Q., Zhang, G., and Badowski, G. (2000). Asymptotic properties of a singularly perturbed Markov chain with inclusion of transient states. Ann. Appl. Probab. 10, 549–572.
Zhang, Q., and Yin, G. (1997). Structural properties of Markov chains with weak and strong interactions. Stochastic Process Appl. 70, 181–197.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yin, G. On Limit Results for a Class of Singularly Perturbed Switching Diffusions. Journal of Theoretical Probability 14, 673–697 (2001). https://doi.org/10.1023/A:1017541022565
Issue Date:
DOI: https://doi.org/10.1023/A:1017541022565