Abstract
We study existence and uniqueness of L^{p} ([0, T] \times \Omega)-bounded mild solutions for a class of semilinear stochastic evolutions equations driven by a general class of Lévy processes without Gaussian component including both the non square integrable (\alpha-stable process) and the square integrable cases on a probability space. This is done using a stochastic analysis on the jumps of the L\'evy process pocess with particular attention to the non square inte-grable case (for instance the α-stable process, \alpha ∈ (0, 2)) through a truncation method by separating the big and small jumps together with a classical fixed point theorem ; under local Lipschitz, Hölder, linear growth conditions on the coefficients. Finally, we give an example to show usefulness of the theoritical results that we obtain in this paper.