TY - JOUR
T1 - Stationary solutions of the stochastic differential equation dVt =Vt -dUt +dLt with Lévy noise
AU - Behme, Anita
AU - Lindner, Alexander
AU - Maller, Ross
PY - 2011/1
Y1 - 2011/1
N2 - For a given bivariate Lvy process (Ut,Lt)t<0, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation dVt=Vt-dUt+dLt are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V0 and (U,L) is assumed and non-causal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for U and L to remain semimartingales with respect to the corresponding expanded filtration is given.
AB - For a given bivariate Lvy process (Ut,Lt)t<0, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation dVt=Vt-dUt+dLt are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V0 and (U,L) is assumed and non-causal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for U and L to remain semimartingales with respect to the corresponding expanded filtration is given.
KW - Filtration expansion
KW - Generalized OrnsteinUhlenbeck process
KW - Lvy process
KW - Non-causal
KW - Stationarity
KW - Stochastic differential equation
KW - Stochastic exponential
UR - http://www.scopus.com/inward/record.url?scp=78149465524&partnerID=8YFLogxK
U2 - 10.1016/j.spa.2010.09.003
DO - 10.1016/j.spa.2010.09.003
M3 - Article
SN - 0304-4149
VL - 121
SP - 91
EP - 108
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
IS - 1
ER -