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A remark on stochastic differential equations with Markov solutions. (Une remarque sur les équations différentielles stochastiques à solutions markoviennes.) (French) Zbl 0741.60051
Séminaire de probabilités, Lect. Notes Math. 1485, 138-139 (1991).
[For the entire collection see Zbl 0733.00018.]
Let $$X$$ be the solution of the stochastic differential equation $dX_ t=f(X_{t-})dZ_ t; \qquad X_ 0=x,$ where $$Z$$ is a semimartingale and $$f$$ is a Borel measurable function such that for each initial condition $$x$$ there is a unique solution $$X^ x$$. It is then well-known that if $$Z$$ is a Lévy process (i.e. $$Z$$ has stationary and independent increments), the processes $$X^ x$$ are all homogeneous Markov processes, with transition semigroups that do not depend on $$x$$. It is shown that this result has a converse: if $$f$$ is never zero and if the processes $$X^ x$$ are all homogeneous Markov with the same transition semigroup, then $$Z$$ must be a Lévy process. A related result is also established when $$Z$$ need only be strong Markov and $$(X^ x,Z)$$ is a vector valued homogeneous Markov process.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)