# Linear diffusion with stationary switching regime

Xavier Guyon; Serge Iovleff; Jian-Feng Yao

ESAIM: Probability and Statistics (2010)

- 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 (2010): 25-35. <http://eudml.org/doc/104320>.

@article{Guyon2010,

abstract = {
Let Y be a Ornstein–Uhlenbeck diffusion governed by a
stationary and ergodic process X : dYt = a(Xt)Ytdt + 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.
},

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; jump process; ergodicity; existence of moments},

language = {eng},

month = {3},

pages = {25-35},

publisher = {EDP Sciences},

title = {Linear diffusion with stationary switching regime},

url = {http://eudml.org/doc/104320},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Guyon, Xavier

AU - Iovleff, Serge

AU - Yao, Jian-Feng

TI - Linear diffusion with stationary switching regime

JO - ESAIM: Probability and Statistics

DA - 2010/3//

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 : dYt = a(Xt)Ytdt + 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.

LA - eng

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

UR - http://eudml.org/doc/104320

ER -

## References

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