We study stochastic Hamilton-Jacobi-Bellman equations and the corresponding Hamiltonian systems driven by jump-type Lévy processes. The main objective of the present paper is to show existence, uniqueness and a (locally in time) diffeomorphism property of the solution: the solution trajectory of the system is a diffeomorphism as a function of the initial impulse. This result enables us to implement a stochastic version of the classical method of characteristics for the Hamilton-Jacobi equations. An -in itself interesting- auxiliary result are pointwise a.s. estimates for iterated stochastic integrals driven by a vector of not necessarily independent jump-type semimartingales.
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Vassili N. Kolokol'tsov, René L. Schilling, Alexei E. Tyukov, Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations. Rev. Mat. Iberoam. 20 (2004), no. 2, pp. 333–380DOI 10.4171/RMI/392