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Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise. (English) Zbl 1168.60014
Let $$H$$ be a separable Hilbert space and $$A:\, D(A)\subset H\rightarrow H$$ be a (in general, unbounded) linear operator generating a pseudo-contraction semigroup. The authors of the present paper are interested in the stochastic differential equation (SDE) $$dZ_t=(AZ_t+B(t,Z))dt+\int_{H\setminus\{0\}}F(t,e,Z)\widetilde{N} (dsde),\, t\in(0,T],\;T>0,$$ where $$Z$$ is an $$H$$-valued process, and $$\widetilde{N} (dsde)=N(dsde)-dsd\beta(du)$$ is a compensated Poisson random measure and $$\beta(de)$$ is a Lévy measure on the Borel-$$\sigma$$-field over $$H\setminus\{0\}$$. They prove the existence and the uniqueness of a mild solution in the case of coefficients $$B(t,.z),F(t,e,z)$$ which can depend on the past of the càdlàd function $$z$$ and are Lipschitz in $$z$$.
While this existence and uniqueness result was obtained for compensated Poisson random measures associated with canonical Lévy processes on the Skorohod space $$D(R_+,H)$$, the authors analyse the case $$B(s,Z)=b(s,Z_s),F(s,e,Z)=f(s,e,Z_s)$$ for more general compensated Poisson random measures. In this case they also prove that a Yosida approximation theorem holds, they study the Markov property and they investigate the continuous and also the differentiable dependence of the solution on the initial data. The latter results are got by adapting a method of S. Cerrai. The authors’ paper generalizes in particular former results by Hausenblas (2005) who studied the same type of SDE, but assumed that the operator $$A$$ generates a compact analytic semigroup.

##### MSC:
 60H05 Stochastic integrals 60G57 Random measures 46B09 Probabilistic methods in Banach space theory 47G99 Integral, integro-differential, and pseudodifferential operators
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