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Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

Tomasz Klimsiak, Andrzej Rozkosz (2016)

Colloquium Mathematicae

We are mainly concerned with equations of the form -Lu = f(x,u) + μ, where L is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, f satisfies the monotonicity condition and mild integrability conditions, and μ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric...

Some results on invariant measures in hydrodynamics

B. Ferrario (2000)

Bollettino dell'Unione Matematica Italiana

In questa nota, si presentano risultati di esistenza e di unicità di misure invarianti per l'equazione di Navier-Stokes che governa il moto di un fluido viscoso incomprimibile omogeneo in un dominio bidimensionale soggetto a una forzante che ha due componenti: una deterministica e una di tipo rumore bianco nella variabile temporale.

Static hedging of barrier options with a smile : an inverse problem

Claude Bardos, Raphaël Douady, Andrei Fursikov (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Let L be a parabolic second order differential operator on the domain Π ¯ = 0 , T × . Given a function u ^ : R and x ^ > 0 such that the support of u ^ is contained in ( - , - x ^ ] , we let y ^ : Π ¯ be the solution to the equation: L y ^ = 0 , y ^ | { 0 } × = u ^ . Given positive bounds 0 < x 0 < x 1 , we seek a function u with support in x 0 , x 1 such that the corresponding solution y satisfies: y ( t , 0 ) = y ^ ( t , 0 ) t 0 , T . We prove in this article that, under some regularity conditions on the coefficients of L , continuous solutions are unique and dense in the sense that y ^ | [ 0 , T ] × { 0 } can be C 0 -approximated, but an exact solution does not...

Static Hedging of Barrier Options with a Smile: An Inverse Problem

Claude Bardos, Raphaël Douady, Andrei Fursikov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Let L be a parabolic second order differential operator on the domain Π ¯ = 0 , T × . Given a function u ^ : R and x ^ > 0 such that the support of û is contained in ( - , - x ^ ] , we let y ^ : Π ¯ be the solution to the equation: L y ^ = 0 , y ^ | { 0 } × = u ^ . Given positive bounds 0 < x 0 < x 1 , we seek a function u with support in x 0 , x 1 such that the corresponding solution y satisfies: y ( t , 0 ) = y ^ ( t , 0 ) t 0 , T . We prove in this article that, under some regularity conditions on the coefficients of L, continuous solutions are unique and dense in the sense that y ^ | [ 0 , T ] × { 0 } can be C0-approximated, but an exact solution...

Stochastic fuzzy differential equations with an application

Marek T. Malinowski, Mariusz Michta (2011)

Kybernetika

In this paper we present the existence and uniqueness of solutions to the stochastic fuzzy differential equations driven by Brownian motion. The continuous dependence on initial condition and stability properties are also established. As an example of application we use some stochastic fuzzy differential equation in a model of population dynamics.

Stochastic Inverse Problem with Noisy Simulator. Application to aeronautical model

Nabil Rachdi, Jean-Claude Fort, Thierry Klein (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

Inverse problem is a current practice in engineering where the goal is to identify parameters from observed data through numerical models. These numerical models, also called Simulators, are built to represent the phenomenon making possible the inference. However, such representation can include some part of variability or commonly called uncertainty (see [4]), arising from some variables of the model. The phenomenon we study is the fuel mass needed to link two given countries with a commercial...

Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions

Mireille Bossy, Mamadou Cissé, Denis Talay (2011)

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

In this paper we explicit the derivative of the flows of one-dimensional reflected diffusion processes. We then get stochastic representations for derivatives of viscosity solutions of one-dimensional semilinear parabolic partial differential equations and parabolic variational inequalities with Neumann boundary conditions.

Stochastic Taylor expansions and heat kernel asymptotics

Fabrice Baudoin (2012)

ESAIM: Probability and Statistics

These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern–Gauss–Bonnet theorem.

Systemic risk through contagion in a core-periphery structured banking network

Oliver Kley, Claudia Klüppelberg, Lukas Reichel (2015)

Banach Center Publications

We contribute to the understanding of how systemic risk arises in a network of credit-interlinked agents. Motivated by empirical studies we formulate a network model which, despite its simplicity, depicts the nature of interbank markets better than a symmetric model. The components of a vector Ornstein-Uhlenbeck process living on the nodes of the network describe the financial robustnesses of the agents. For this system, we prove a LLN for growing network size leading to a propagation of chaos result....

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