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Incompleteness of the bond market with Lévy noise under the physical measure

Michał Barski (2015)

Banach Center Publications

The problem of completeness of the forward rate based bond market model driven by a Lévy process under the physical measure is examined. The incompleteness of market in the case when the Lévy measure has a density function is shown. The required elements of the theory of stochastic integration over the compensated jump measure under a martingale measure are presented and the corresponding integral representation of local martingales is proven.

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...

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