# Linear diffusion with stationary switching regime

Xavier Guyon; Serge Iovleff; Jian-Feng Yao

ESAIM: Probability and Statistics (2004)

- Volume: 8, page 25-35
- ISSN: 1292-8100

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topGuyon, Xavier, Iovleff, Serge, and Yao, Jian-Feng. "Linear diffusion with stationary switching regime." ESAIM: Probability and Statistics 8 (2004): 25-35. <http://eudml.org/doc/244728>.

@article{Guyon2004,

abstract = {Let $Y$ be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process $X : \{\rm d\}Y_\{t\}= a(X_\{t\})Y_\{t\}\{\rm d\}t+\sigma (X_\{t\})\{\rm d\}W_\{t\},Y_\{0\}=y_\{0\}$. We establish that under the condition $\alpha =E_\{\mu \}(a(X_\{0\}))<0$ with $\mu $ 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 $s\ge 0$. 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.},

author = {Guyon, Xavier, Iovleff, Serge, Yao, Jian-Feng},

journal = {ESAIM: Probability and Statistics},

keywords = {Ornstein–Uhlenbeck diffusion; Markov switching; jump process; random difference equations; ergodicity; existence of moments; Ornstein-Uhlenbeck diffusion},

language = {eng},

pages = {25-35},

publisher = {EDP-Sciences},

title = {Linear diffusion with stationary switching regime},

url = {http://eudml.org/doc/244728},

volume = {8},

year = {2004},

}

TY - JOUR

AU - Guyon, Xavier

AU - Iovleff, Serge

AU - Yao, Jian-Feng

TI - Linear diffusion with stationary switching regime

JO - ESAIM: Probability and Statistics

PY - 2004

PB - EDP-Sciences

VL - 8

SP - 25

EP - 35

AB - Let $Y$ be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process $X : {\rm d}Y_{t}= a(X_{t})Y_{t}{\rm d}t+\sigma (X_{t}){\rm d}W_{t},Y_{0}=y_{0}$. We establish that under the condition $\alpha =E_{\mu }(a(X_{0}))<0$ with $\mu $ 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 $s\ge 0$. 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.

LA - eng

KW - Ornstein–Uhlenbeck diffusion; Markov switching; jump process; random difference equations; ergodicity; existence of moments; Ornstein-Uhlenbeck diffusion

UR - http://eudml.org/doc/244728

ER -

## References

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