Revista Matemática Iberoamericana

Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations

Vassili N. Kolokol'tsov, René L. Schilling, and Alexei E. Tyukov

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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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Rev. Mat. Iberoamericana, Volume 20, Number 2 (2004), 333-380.

First available in Project Euclid: 17 June 2004

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Zentralblatt MATH identifier

Primary: 70H20: Hamilton-Jacobi equations 60H15: Stochastic partial differential equations [See also 35R60] 60H05: Stochastic integrals
Secondary: 60G51: Processes with independent increments; Lévy processes 60J75: Jump processes 70H05: Hamilton's equations

stochastic Hamilton-Jacobi equation Hamiltonian system method of stochastic characteristics iterated stochastic integral semimartingale Lévy process


Kolokol'tsov, Vassili N.; Schilling, René L.; Tyukov, Alexei E. Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations. Rev. Mat. Iberoamericana 20 (2004), no. 2, 333--380.

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