Appl Math Optim 40:105–140 (1999)
1999 Springer-Verlag New York Inc.
Invariant Measure for Diffusions with Jumps
and M. Robin
Department of Mathematics, Wayne State University,
MI 48202, USA
European Organization for Nuclear Research, CERN,
Geneva 23, Switzerland CH-1211
Communicated by A. Bensoussan
Abstract. Ourpurpose is tostudy an ergodiclinearequationassociated todiffusion
processes with jumps in the whole space. This integro-differential equation plays a
fundamentalrole inergodiccontrol problemsof secondorder Markovprocesses.The
keyresult is to prove the existence and uniqueness of an invariantdensity functionfor
a jump diffusion, whose lower order coefﬁcients are only Borel measurable. Based
on this invariant probability, existence and uniqueness (up to an additive constant)
of solutions to the ergodic linear equation are established.
Key Words. Jump diffusion, Interior Dirichlet problem, Exterior Dirichlet prob-
lem, Ergodic optimal control, Green function, Girsanov transformation, Doeblin
AMS Classiﬁcation. Primary 35J25, 60J60, 60J75, Secondary 45K05, 46N20,
49A60, 93D05, 93E20.
Ergodic properties of diffusion processes and its relation with partial differential equa-
tions are well know in the classic literature. However, similar questions for diffusion
processes with jumps are not so popular, only recently was some attention given, see
, , and the reference therein.