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Zbl 0741.60051
Jacod, J.; Protter, P.
A remark on stochastic differential equations with Markov solutions. (Une remarque sur les équations différentielles stochastiques à solutions markoviennes.) (French)
[A] Séminaire de probabilités, Lect. Notes Math. 1485, 138-139 (1991).

[For the entire collection see Zbl 0733.00018.]\par Let $X$ be the solution of the stochastic differential equation $$dX\sb t=f(X\sb{t-})dZ\sb t; \qquad X\sb 0=x,$$ where $Z$ is a semimartingale and $f$ is a Borel measurable function such that for each initial condition $x$ there is a unique solution $X\sp x$. It is then well-known that if $Z$ is a Lévy process (i.e. $Z$ has stationary and independent increments), the processes $X\sp x$ 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 $X\sp x$ 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 $(X\sp x,Z)$ is a vector valued homogeneous Markov process.
[P.Protter (West Lafayette)]

MSC 2000:: *60H10 Stochastic ordinary differential equations

Keywords: semimartingales; stochastich differential equation; Lévy process; transition semigroup; strong Markov

Citations: Zbl 0733.00018

Cited in: Zbl 0889.60066

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