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In this work we consider a finite dimensional stochastic differential equation(SDE) driven by a Lévy noise L (t) = L t , t > 0. The transition probability density p t , t > 0 of the semigroup associated to the solution u t , t ⩾ 0 of the SDE is given by a power series expansion. The series expansion of p t can be re-expressed in terms of Feynman graphs and rules. We will also prove that p t , t > 0 has an asymptotic expansion in power of a parameter β > 0 , and it can be given by a convergent integral. A remark on some applications will be given in this work. [ABSTRACT FROM AUTHOR] more...