Revista Matemática Iberoamericana

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Volume 20, Issue 2, 2004, pp. 333–380
DOI: 10.4171/RMI/392

Published online: 2004-08-31

Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations

Vassili N. Kolokol'tsov[1], René L. Schilling[2] and Alexei E. Tyukov[3]

(1) Nottingham Trent University, UK
(2) Technische Universität Dresden, Germany
(3) Cardiff University, UK

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.

Keywords: Stochastic Hamilton-Jacobi equation, Hamiltonian system, method of stochastic characteristics, iterated stochastic integral, semimartingale, Lévy process

Kolokol'tsov Vassili, Schilling René, Tyukov Alexei: Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations. Rev. Mat. Iberoam. 20 (2004), 333-380. doi: 10.4171/RMI/392