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Linear diffusion with stationary switching regime

  • Xavier Guyon (a1), Serge Iovleff (a2) and Jian-Feng Yao (a3)
Abstract

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process X : dYt = a(Xt)Yt dt + σ(Xt)dWt,Y0 = y0 . We establish that under the condition α = Eµ(a(X0)) < 0 with μ the stationary distribution of the regime process X, the diffusion Y is ergodic. We also consider conditions for the existence of moments for the invariant law of Y when X is a Markov jump process having a finite number of states. Using results on random difference equations on one hand and the fact that conditionally to X, Y is Gaussian on the other hand, we give such a condition for the existence of the moment of order s ≥ 0. Actually we recover in this case a result that Basak et al. [J. Math. Anal. Appl. 202 (1996) 604–622] have established using the theory of stochastic control of linear systems.

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References
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[1] Basak, G.K., Bisi, A. and Ghosh, M.K., Stability of random diffusion with linear drift. J. Math. Anal. Appl. 202 (1996) 604-622.
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ESAIM: Probability and Statistics
  • ISSN: 1292-8100
  • EISSN: 1262-3318
  • URL: /core/journals/esaim-probability-and-statistics
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