# American Institute of Mathematical Sciences

August  2019, 24(8): 3615-3631. doi: 10.3934/dcdsb.2018307

## Smoothness of density for stochastic differential equations with Markovian switching

 1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 2 Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045, USA 3 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

* Corresponding author

Received  December 2017 Revised  May 2018 Published  August 2019 Early access  November 2018

Fund Project: Y. Hu is partially supported by a grant from the Simons Foundation #209206. D. Nualart is supported by the NSF grant DMS1512891. X. Sun and Y. Xie are supported by Natural Science Foundation of China (11601196, 11771187), Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (16KJB110006) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

This paper is concerned with a class of stochastic differential equations with Markovian switching. The Malliavin calculus is used to study the smoothness of the density of the solution under a Hörmander type condition. Furthermore, we obtain a Bismut type formula which is used to establish the strong Feller property.

Citation: Yaozhong Hu, David Nualart, Xiaobin Sun, Yingchao Xie. Smoothness of density for stochastic differential equations with Markovian switching. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3615-3631. doi: 10.3934/dcdsb.2018307
##### References:
 [1] G. Basak, A. Bisi and M. Ghosh, Stability of a Random Diffusion with Linear Drift, J. Math. Anal. Appl., 202 (1996), 604-622.  doi: 10.1006/jmaa.1996.0336. [2] B. Forster, E. Lütkebohmert and J. Teichmann, Absolutely continuous laws of jump-diffusions in finite and infinite dimensions with appliationc to mathematical finance, SIAM J. Math. Anal., 40 (2009), 2132-2153.  doi: 10.1137/070708822. [3] P. Malliavin, Stochastic Analysis, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-15074-6. [4] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2. [5] D. Nualart, The Malliavin Calculus and Related Topics, Springer, Berlin, 2006. [6] G. Yin and C. Zhu,, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [7] C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 103 (2003), 277-291.  doi: 10.1016/S0304-4149(02)00230-2.

show all references

##### References:
 [1] G. Basak, A. Bisi and M. Ghosh, Stability of a Random Diffusion with Linear Drift, J. Math. Anal. Appl., 202 (1996), 604-622.  doi: 10.1006/jmaa.1996.0336. [2] B. Forster, E. Lütkebohmert and J. Teichmann, Absolutely continuous laws of jump-diffusions in finite and infinite dimensions with appliationc to mathematical finance, SIAM J. Math. Anal., 40 (2009), 2132-2153.  doi: 10.1137/070708822. [3] P. Malliavin, Stochastic Analysis, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-15074-6. [4] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2. [5] D. Nualart, The Malliavin Calculus and Related Topics, Springer, Berlin, 2006. [6] G. Yin and C. Zhu,, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [7] C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 103 (2003), 277-291.  doi: 10.1016/S0304-4149(02)00230-2.
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