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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.

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