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Theory and applications of Lévy processes (stochastic processes with stationary and independent increments): e.g. path properties, stochastic differential equations driven by jump-type processes, fluctuation theory of Lévy processes, queuing theory.
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Lévy measure and Lévy process
No, for a counterexample, just take
$$
\nu(dx) = \left(C_1 x^{-2} I(0<x<1) +C_2 |x|^{-2} I(-1<x<0)\right)dx,
$$
where $C_1\neq C_2$. Then,
$$
\int_{r<|x|\le 1} x\nu(dx) = (C_2-C_1) \int_r^1 x^{-1} dx …