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Intégrales stochastiques de processus anticipants et projections duales prévisibles.

Catherine Donati-Martin, Marc Yor (1999)

Publicacions Matemàtiques

We define a stochastic anticipating integral δμ with respect to Brownian motion, associated to a non adapted increasing process (μt), with dual projection t. The integral δμ(u) of an anticipating process (ut) satisfies: for every bounded predictable process ft,E [ (∫ fsdBs ) δμ(u) ] = E [ ∫ fsusdμs ].We characterize this integral when μt = supt ≤s ≤ 1 Bs. The proof relies on a path decomposition of Brownian motion up to time 1.

Integration in a dynamical stochastic geometric framework

Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso (2011)

ESAIM: Probability and Statistics

Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary...

Integration in a dynamical stochastic geometric framework

Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso (2012)

ESAIM: Probability and Statistics

Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary...

Intermittency properties in a hyperbolic Anderson problem

Robert C. Dalang, Carl Mueller (2009)

Annales de l'I.H.P. Probabilités et statistiques

We study the asymptotics of the even moments of solutions to a stochastic wave equation in spatial dimension 3 with linear multiplicative spatially homogeneous gaussian noise that is white in time. Our main theorem states that these moments grow more quickly than one might expect. This phenomenon is well known for parabolic stochastic partial differential equations, under the name of intermittency. Our results seem to be the first example of this phenomenon for hyperbolic equations. For comparison,...

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