Abstract
This work develops asymptotic expansions of systems of partial differential equations associated with multi-scale switching diffusions. The switching process is modeled by using an inhomogeneous continuous-time Markov chain. In the model, there are two small parameters ε and δ. The first one highlights the fast switching, whereas the other delineates the slow diffusion. Assuming that ε and δ are related in that ε = δγ, our results reveal that different values of γ lead to different behaviors of the underlying systems, resulting in different asymptotic expansions. Although our motivation comes from stochastic problems, the approach is mainly analytic and is constructive. The asymptotic series are rigorously justified with error bounds provided. An example is provided to demonstrate the results.
Similar content being viewed by others
References
Eidelman, S.D. Parabolic Systems. North-Holland, New York, 1969
Fleming, W.H. Stochastically perturbed dynamical systems. Rocky Mountain J. Math. 4: 407–433 (1974)
Friedman, A. Partial differential equations of parabolic type. Courier Corporation, Mineola, New York, 2013
He, Q., Yin, G. Large deviations for multi-scale Markovian switching systems with a small diffusion. Asymptotic Anal. 87 (3-4): 123–145 (2014)
Holland, C.J. Ergodic expansions in small noise problems. J. Differential Equations, 16: 281–288 (1974)
Ilin, A.M. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Trans. Math. Monographs. V. 102. Amer. Math. Soc., Providence, 1992
Ilin, A.M., Khasminskii, R.Z., Yin, G. Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: rapid switchings. J. Math. Anal. Appl. 238(2): 516–539 (1999)
Tran, K., Yin, G., Wang, L.Y., Zhang, H. Singularly perturbed multi-scale switching diffusions. Dynamic Systems and Appl., 25(1-2): 153–174 (2016)
Khasminskii, R.Z., Yin, G. Uniform asymptotic expansions for pricing European options. Appl. Math. Optim., 52: 279–296 (2005)
Khasminskii, R.Z., Yin, G. Limit behavior of two-time-scale diffusions revisited. J. Differential Eqs., 212: 85–113 (2005)
Massey, W.A., Whitt, W. Uniform acceleration expansions for Markov chains with time-varying rates. Ann. Appl. Probab., 8: 1130–1155 (1998)
Nguyen, D., Yin, G. Asymptotic expansions of solutions of systems of Kolmogorov backward equations for two-time-scale switching diffusions. Quart. Appl. Math., 71(4): 601–628 (2013)
Polyanin, A.D., Zaitsev, V.F., Moussiaux, A. Handbook of First Order Partial Differential Equations. Taylor & Francis, London, 2002
Sethi, S.P., Zhang, H., Zhang, Q. Average-Cost Control of Stochastic Manufacturing Systems. Springer- Verlag, New York, NY, 2004
Sethi, S.P., Zhang, Q. Hierarchical decision making in stochastic manufacturing systems. Systems & Control: Foundations & Applications. Birkhuser Boston, Inc., Boston, MA, 1994
Simon, H.A., Ando, A. Aggregation of variables in dynamic systems. Econometrica, 29: 111–138 (1961)
Tsai, C-P. Perturbed linear stochastic dynamical systems. J. Differential Eqs., 50(1): 146–161 (1983)
Yin, G., Zhang, Q. Continuous-time Markov chains and applications. A two-time-scale approach, 2nd Ed., Springer, New York, 2013
Yin, G., Zhu, C. Properties of solutions of stochastic differential equations with continuous-state-dependent switching. J. Differential Eqs., 249(10): 2409–2439 (2010)
Yin, G., Zhu, C. Hybrid Switching Diffusions: Properties and Applications. Springer, New York, 2010
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported in part by the Air Force Office of Scientific Research under FA9550-15-1-0131.
Rights and permissions
About this article
Cite this article
Tran, K., Yin, G. Asymptotic expansions for solutions of parabolic systems associated with multi-scale switching diffusions. Acta Math. Appl. Sin. Engl. Ser. 33, 731–752 (2017). https://doi.org/10.1007/s10255-017-0695-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-017-0695-9