Almost sure stability of linear stochastic differential equations with jumps |
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Authors: | C W Li Z Dong R Situ |
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Institution: | (1) Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong. e-mail: macwli@cityu.edu.hk (corresponding author), HK;(2) Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing 100080, P. R. China., CN;(3) Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China, CN |
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Abstract: | Under the nondegenerate condition as in the diffusion case, see 14, 21, 6], the linear stochastic jump-diffusion process
projected on the unit sphere is a strong Feller process and has a unique invariant measure which is also ergodic using the
relation between the transition probabilities of jump-diffusions and the corresponding diffusions due to 22]. The unique
deterministic Lyapunov exponent can be represented by the Furstenberg-Khas'minskii formula as an integral over the sphere
or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear
stochastic systems with jumps depends on its sign. The critical case of zero Lyapunov exponent is discussed and a large deviations
result for asymptotically stable systems is further investigated. Several examples are treated for illustration.
Received: 22 June 2000 / Revised version: 20 November 2001 / Published online: 13 May 2002 |
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