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In this informal talk, we propose to review some basic models of stochastic evolution in time and we discuss two different problems both consisting in the computation of functions defined along the trajectories of this processes.

Part I will be devoted to the presentation of the Brownian Motion, the Poisson process, the compound Poisson processes, and the Lévy processes.

In part II of the talk we plan to present some exact results for ruin probabilities for Levy processes.

In part III we generalize the class of stochastic processes considered, by defining stochastic differential equations driven by processes with jumps.

Finally, in part IV, we present some recent results on weak approximation of diffusion with jumps (this is joint work with A. Szepessy, R. Tempone and G. Zouraris), and discuss some possible further generalization of this work.

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