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95
Construction of Diffusions on Configuration Spaces
"... We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Ga ..."
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Cited by 27 (3 self)
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We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Gamma E ; ¯) for a large class of measures ¯ on \Gamma E (without assuming an integration by parts formula) generalizing all corresponding results known so far. Subsequently, we prove that under mild conditions the Dirichlet forms E \Gamma ¯ are quasiregular, and that hence there exist associated diffusions on \Gamma E , provided E is a complete separable metric space and \Gamma E is equipped with a suitable topology, which is the vague topology if E is locally compact. We discuss applications to the case where E is a finite dimensional manifold yielding an existence result on diffusions on \Gamma E which was already announced in [AKR96a, AKR96b], resp. used in [AKR98, AKR97b]. Furthermore...
A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
"... The aim of this work is to evaluate the cheapest superreplication price of a general (possibly pathdependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. Th ..."
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Cited by 17 (0 self)
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The aim of this work is to evaluate the cheapest superreplication price of a general (possibly pathdependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case. 1. Introduction. Our
Raimond O.: Integration of Brownian vector fields
 Annals of Prob
"... Summary. Using the Wiener chaos decomposition, we show that strong solutions of non Lipschitzian S.D.E.’s are given by random Markovian kernels. The example of Sobolev flows is studied in some detail, exhibiting interesting phase transitions. ..."
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Cited by 17 (2 self)
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Summary. Using the Wiener chaos decomposition, we show that strong solutions of non Lipschitzian S.D.E.’s are given by random Markovian kernels. The example of Sobolev flows is studied in some detail, exhibiting interesting phase transitions.
Equivalence of Gradients on Configuration Spaces
 Random Operators and Stochastic Equations
, 1999
"... The gradient on a Riemannian manifold X is lifted to the configuration space \Upsilon X on X via a pointwise identity. This entails a norm equivalence that either holds under any probability measure or characterizes the Poisson measures, depending on the tangent space chosen on \Upsilon X . More ..."
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Cited by 14 (4 self)
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The gradient on a Riemannian manifold X is lifted to the configuration space \Upsilon X on X via a pointwise identity. This entails a norm equivalence that either holds under any probability measure or characterizes the Poisson measures, depending on the tangent space chosen on \Upsilon X . More generally, this approach links carr'e du champ operators on X to their counterparts on \Upsilon X , and also includes structures that do not admit a gradient. Key words: Configuration spaces, Poisson measures, Stochastic analysis. Mathematics Subject Classification (1991): 58G32, 60H07, 60J45, 60J75. 1 Introduction Stochastic analysis under Poisson measures, cf. [5], [6], has been developed in several different directions. This is mainly due to the fact that, unlike on the Wiener space, the gradient on Fock space and the infinitesimal Poisson gradient do not coincide under the identification of the Fock space to the L 2 space of the Poisson process.  The gradient on Fock space is in...
A duality approach for the weak approximation of stochastic differential equations, in "Annals of Applied Probability", vol. 16, n o 3
, 2006
"... In this article we develop a new methodology to prove weak approximation results for general stochastic differential equations. Instead of using a partial differential equation approach as is usually done for diffusions, the approach considered here uses the properties of the linear equation satisfi ..."
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Cited by 13 (0 self)
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In this article we develop a new methodology to prove weak approximation results for general stochastic differential equations. Instead of using a partial differential equation approach as is usually done for diffusions, the approach considered here uses the properties of the linear equation satisfied by the error process. This methodology seems to apply to a large class of processes and we present as an example the weak approximation of stochastic delay equations. 1. Introduction. The Euler
Ergodicity for the stochastic dynamics of quasiinvariant measures with applications to Gibbs states
, 1997
"... The convex set M a of quasiinvariant measures on a locally convex space E with given "shift"RadonNikodym derivatives (i.e., cocycles) a = (a tk ) k2K 0 ; t2R is analyzed. The extreme points of M a are characterized and proved to be nonempty. A specification (of lattice type) is constructed ..."
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Cited by 10 (2 self)
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The convex set M a of quasiinvariant measures on a locally convex space E with given "shift"RadonNikodym derivatives (i.e., cocycles) a = (a tk ) k2K 0 ; t2R is analyzed. The extreme points of M a are characterized and proved to be nonempty. A specification (of lattice type) is constructed so that M a coincides with the set of the corresponding Gibbs states. As a consequence, via a wellknown method due to DynkinFollmer a unique representation of an arbitrary element in M a in terms of extreme ones is derived. Furthermore, the corresponding classical Dirichlet forms (E ; D(E )) and their associated semigroups (T t ) t?0 on L 2 (E; ) are discussed. Under a mild positivity condition it is shown that 2 M a is extreme if and only if (E ; D(E )) is irreducible or equivalently, (T t ) t?0 is ergodic. This implies timeergodicity of associated diffusions. Applications to Gibbs states of classical and quantum lattice models as well as those occuring in Euclidean...
(Nonsymmetric) Dirichlet Operators On L¹: Existence, Uniqueness And Associated Markov Processes
"... Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitel ..."
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Cited by 9 (2 self)
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Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitely construct, under mild regularity assumptions, extensions of L generating subMarkovian C0 semigroups on L 1 (U; ¯) as well as associated diffusion processes. We give sufficient conditions on the coefficients so that there exists only one extension of L generating a C0 semigroup and apply the results to prove uniqueness of the invariant measure ¯. Our results imply in particular that if ' 2 H 1;2 loc (R d ; dx), ' 6= 0 dxa.e., the generalized Schrödinger operator (\Delta + 2' \Gamma1 r' \Delta r;C 1 0 (R d )) has exactly one extension generating a C0 semigroup if and only if the Friedrich's extension is conservative. We also give existence and uniqueness results for ...
Error calculus and path sensitivity in Financial models”, Mathematical Finance vol 13/1
, 2003
"... In the framework of risk management, for the study of the sensitivity of pricing and hedging in stochastic financial models to changes of parameters and to perturbations of the stock prices, we propose an error calculus which is an extension of the Malliavin calculus based on Dirichlet forms. Althou ..."
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Cited by 9 (7 self)
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In the framework of risk management, for the study of the sensitivity of pricing and hedging in stochastic financial models to changes of parameters and to perturbations of the stock prices, we propose an error calculus which is an extension of the Malliavin calculus based on Dirichlet forms. Although useful also in physics, this error calculus is well adapted to stochastic analysis and seems to be the best practicable in finance. This tool is explained here intuitively and with some simple examples.
Harmonic coordinates on fractals with finitely ramified cell structure
 Department of Mathematics, University of Copenhagen, DK2100 Copenhagen, Denmark Email address: echris@math.ku.dk Department of Mathematics, University of Hannover
"... Abstract. We define sets with finitely ramified cell structure, which are generalizations of p.c.f. selfsimilar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local selfsimilarity, and allow countably many cells connected at each junction p ..."
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Cited by 8 (4 self)
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Abstract. We define sets with finitely ramified cell structure, which are generalizations of p.c.f. selfsimilar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local selfsimilarity, and allow countably many cells connected at each junction point. In particular, we consider postcritically infinite fractals. We prove that if Kigami’s resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates. Contents