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**11 - 18**of**18**### A Note on Existence and Path Properties of Space-Time Processes corresponding to Time Dependent Fleming-Viot Operators

, 1999

"... We study a class of time dependent Dirichlet forms (E, D(E)) having a space of measures as state space E and derive some general results about them. Especially we prove that under certain conditions they "generate" space-time diffusion processes. All of this is applied to construct time-dependent Fl ..."

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We study a class of time dependent Dirichlet forms (E, D(E)) having a space of measures as state space E and derive some general results about them. Especially we prove that under certain conditions they "generate" space-time diffusion processes. All of this is applied to construct time-dependent Fleming-Viot processes and derive some path properties using parabolic capacitary methods. We establish a general method to compare the parabolic and the elliptic case.

### Some Conditions for the Existence of Singular Non Symmetric Diffusions

"... Some new conditions which ensure the existence of diusion processes with values in R properly associated to NonSymmetric Dirichlet Forms are given. The results are extended to the case of diusions taking values in Wiener Spaces. ..."

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Some new conditions which ensure the existence of diusion processes with values in R properly associated to NonSymmetric Dirichlet Forms are given. The results are extended to the case of diusions taking values in Wiener Spaces.

### Generalized covariation and extended Fukushima decompositions for Banach valued processes. Application to windows of Dirichlet processes.

, 2011

"... This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of Métivier and Pellaumail which is quite restrictive. We make use of the notion of χ-covariation which is a generalized notion of covariation for p ..."

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This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of Métivier and Pellaumail which is quite restrictive. We make use of the notion of χ-covariation which is a generalized notion of covariation for processes with values in two Banach spaces B1 and B2. χ refers to a suitable subspace of the dual of the projective tensor product of B1 and B2. We investigate some C 1 type transformations for various classes of stochastic processes admitting a χ-quadratic variation and related properties. If X 1 and X 2 admit a χ-covariation, F i: Bi → R, i = 1,2 are of class C 1 with some supplementary assumptions then the covariation of the real processes F 1 (X 1) and F 2 (X 2) exist. A detailed analysis will be devoted to the so-called window processes. Let X be a real continuous process; the C([−τ,0])-valued process X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ,0], is called window process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. In fact we aim to generalize the following properties valid for B = R. If X = X is a real valued Dirichlet process and F: B → R of class C 1 (B) then F(X) is still a Dirichlet process. If X = X is a weak Dirichlet process with finite quadratic variation,

### DIRICHLET FORMS METHODS: AN APPLICATION TO THE PROPAGATION OF THE ERROR DUE TO THE EULER SCHEME

, 2006

"... We present recent advances on Dirichlet forms methods either to extend financial models beyond the usual stochastic calculus or to study stochastic models with less classical tools. In this spirit, we interpret the asymptotic error on the solution of an sde due to the Euler scheme (Kurtz and Protter ..."

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We present recent advances on Dirichlet forms methods either to extend financial models beyond the usual stochastic calculus or to study stochastic models with less classical tools. In this spirit, we interpret the asymptotic error on the solution of an sde due to the Euler scheme (Kurtz and Protter [Ku-Pr-91a]) in terms of a Dirichlet form on the Wiener space, what allows to propagate this error thanks to functional calculus.

### Discipline: Mathématiques

, 2011

"... pour le grade de DOCTEUR de per il titolo di DOTTORE DI RICERCA l’Université de Paris 13 dell ’ Università LUISS GUIDO CARLI ..."

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pour le grade de DOCTEUR de per il titolo di DOTTORE DI RICERCA l’Université de Paris 13 dell ’ Università LUISS GUIDO CARLI

### √ − b

, 804

"... Weak existence of the squared Bessel process, CIR model, and Longstaff model, with skew reflection on a deterministic time dependent curve 1 ..."

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Weak existence of the squared Bessel process, CIR model, and Longstaff model, with skew reflection on a deterministic time dependent curve 1

### √ − b

, 804

"... Weak existence of the squared Bessel and CIR process with skew reflection on a deterministic time dependent curve ..."

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Weak existence of the squared Bessel and CIR process with skew reflection on a deterministic time dependent curve

### [u]

"... Summary. Let E be a Hausdorff space such that its Borel σ-algebra is generated by the set of all continuous functions on E. Given a conservative generalized Dirichlet form E with associated diffusion (Xt, Px), reference measure m, and domain F, on L2 (E; m). Suppose that the co-form with domain ˆ F ..."

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Summary. Let E be a Hausdorff space such that its Borel σ-algebra is generated by the set of all continuous functions on E. Given a conservative generalized Dirichlet form E with associated diffusion (Xt, Px), reference measure m, and domain F, on L2 (E; m). Suppose that the co-form with domain ˆ F is also conservative and associated to a diffusion.