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On Backward Stochastic Differential Equations Approach to Valuation of American Options

100%
Klimsiak T., Rozkosz A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2011
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tom 59
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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.
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Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

100%
Klimsiak T., Rozkosz A.
Colloquium Mathematicum
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2016
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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.
3
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Nonlinear parabolic SPDEs involving Dirichlet operators

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Klimsiak T., Rozkosz A.
Studia Mathematica
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2015
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tom 230
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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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