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On mild solutions of gradient systems in Hilbert spaces

100%
Rozkosz A.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1994-2004
EN
We consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.
2
Content available remote

BSDEs with random terminal time and semilinear elliptic PDEs in divergence form

100%
Rozkosz A.
Studia Mathematica
|
2005
|
tom 170
|
nr 1
1-21
EN
We study connections between Sobolev space solutions of the Dirichlet problem for semilinear second order elliptic equations in divergence form and solutions of backward stochastic differential equations with random terminal time.
3
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On Backward Stochastic Differential Equations Approach to Valuation of American Options

64%
Klimsiak T., Rozkosz A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
tom 59
|
nr 3
275-288
EN
We consider the problem of valuation of American (call and put) options written on a dividend paying stock governed by the geometric Brownian motion. We show that the value function has two different but related representations: by means of a solution of some nonlinear backward stochastic differential equation, and by a weak solution to some semilinear partial differential equation.
4
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Stochastic representation of reflecting diffusions corresponding to divergence form operators

64%
Rozkosz A., Słomiński L.
Studia Mathematica
|
2000
|
tom 139
|
nr 2
141-174
EN
We obtain a stochastic representation of a diffusion corresponding to a uniformly elliptic divergence form operator with co-normal reflection at the boundary of a bounded $C^2$-domain. We also show that the diffusion is a Dirichlet process for each starting point inside the domain.
5
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Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

64%
Klimsiak T., Rozkosz A.
Colloquium Mathematicum
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2016
|
tom 145
|
nr 1
35-67
EN
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 divergence form operators, with gradient perturbations of some pseudodifferential operators and equations with Ornstein-Uhlenbeck type operators in Hilbert spaces. We also briefly discuss the existence and uniqueness of probabilistic solutions in the case where L corresponds to a lower bounded semi-Dirichlet form.
6
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Nonlinear parabolic SPDEs involving Dirichlet operators

64%
Klimsiak T., Rozkosz A.
Studia Mathematica
|
2015
|
tom 230
|
nr 3
215-263
EN
We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet forms. In the proofs we combine the methods of backward doubly stochastic differential equations with those of probabilistic potential theory and Dirichlet forms.
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