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

Abstract

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Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process X : d Y t = a ( X t ) Y t d t + σ ( X t ) d W t , Y 0 = y 0 . We establish that under the condition α = E μ ( a ( X 0 ) ) < 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.

How to cite

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Guyon, 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 $\alpha =E_\{\mu \}(a(X_\{0\}))&lt;0$ 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 $\alpha =E_{\mu }(a(X_{0}))&lt;0$ 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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