arXiv subjects

Tomasz Klimsiak

Publications and source records attributed to Tomasz Klimsiak.

At least 55 records · Page 3Linked to original sources

arXiv2017

Trace operator and the Dirichlet problem for elliptic equations on arbitrary bounded open sets

We consider the Dirichlet problem on general, possibly nonsmooth bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, soft and the Perron-Wiener-Brelot solutions of the problem. To this end, we extend the usual notion of the trace operator to Sobolev space $H^1(D)$ with $D$ being an arbitrary bounded open subset of $\mathbb{R}^d$. In the second part of the paper, we prove some existence results for the Dirichlet problem for semilinear equations with measure data on the right-hand side and $L^1$-data on the Martin boundary of $D$.

math.AP↗

arXiv2017

The valuation of American options in a multidimensional exponential Levy model

We consider the problem of valuation of American options written on dividend-paying assets whose price dynamics follows a multidimensional exponential Levy model. We carefully examine the relation between the option prices, related partial integro-differential variational inequalities and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for broad class of payoff functions.

math.PR↗

arXiv2017

On the structure of bounded smooth measures associated with a quasi-regular Dirichlet form

We consider a quasi-regular Dirichlet form. We show that a bounded signed measure charges no set of zero capacity associated with the form if and only if the measure can be decomposed into the sum of an integrable function and a bounded linear functional on the domain of the form. The decomposition allows one to describe explicitly the set of bounded measures charging no sets of zero capacity for interesting classes of Dirichlet forms. By way of illustration, some examples are given.

math.FA↗

arXiv2016

Semilinear elliptic equations with Dirichlet operator and singular nonlinearities

In the paper we consider elliptic equations of the form $-Au=u^{-γ}\cdotμ$, where $A$ is the operator associated with a regular symmetric Dirichlet form, $μ$ is a positive nontrivial measure and $γ>0$. We prove the existence and uniqueness of solutions of such equations as well as some regularity results. We also study stability of solutions with respect to the convergence of measures on the right-hand side of the equation. For this purpose, we introduce some type of functional convergence of smooth measures, which in fact is equivalent to the quasi-uniform convergence of associated potentials.

math.AP↗

arXiv2016

Reduced measures for semilinear elliptic equations involving Dirichlet operators

We consider elliptic equations of the form (E) $-Au=f(x,u)+μ$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $μ$ is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where $A=Δ$ and show basic properties of solutions of (E). We also prove Kato's type inequality. Finally, we characterize the set of good measures in case $f(u)=-u^p$ for some $p>1$.

math.AP↗

arXiv2016

Systems of quasi-variational inequalities related to the switching problem

We prove the existence of weak solution for a system of quasi-variational inequalities related to a switching problem with dynamic driven by operator associated with a semi-Dirichlet form and with measure data. We give a stochastic representation of solutions in terms of solutions of a system of reflected BSDEs with oblique reflection. As a by-product, we prove the existence of an optimal strategy in the switching problem and show regularity of the payoff function.

math.PR↗

arXiv2016

Systems of semilinear parabolic variational inequalities with time-dependent convex obstacles

We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations.

math.AP↗

arXiv2016

Reflected BSDEs with regulated trajectories

We consider reflected backward stochastic different equations with optional barrier and so-called regulated trajectories, i.e trajectories with left and right finite limits. We prove existence and uniqueness results. We also show that the solution may be approximated by a modified penalization method. Application to an optimal stopping problem is given.

math.PR↗

arXiv2016

Nonlinear parabolic SPDEs involving Dirichlet operators

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 proofs we combine the methods of backward doubly stochastic differential equations with those of probabilistic potential theory and Dirichlet forms.

math.PR↗

arXiv2016

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

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.

math.AP↗

arXiv2015

Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form

We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous function $u$ is a renormalized solution to an elliptic (or parabolic) equation in the sense of our definition iff $u$ is its probabilistic solution, i.e. $u$ can be represented by a suitable nonlinear Feynman-Kac formula. This implies in particular that for a broad class of local and nonlocal semilinear equations there exists a unique renormalized solution.

math.AP↗

arXiv2015

Right Markov processes and systems of semilinear equations with measure data

In the paper we prove the existence of probabilistic solutions to systems of the form $-Au=F(x,u)+μ$, where $F$ satisfies a generalized sign condition and $μ$ is a smooth measure. As for $A$ we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on $L^1$. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer's property (L) of Markov processes and in terms of order compactness of the associated resolvent.

math.AP↗

arXiv2015

The early exercise premium representation for American options on multiply assets

In the paper we consider the problem of valuation of American options written on dividend-paying assets whose price dynamics follow the classical multidimensional Black and Scholes model. We provide a general early exercise premium representation formula for options with payoff functions which are convex or satisfy mild regularity assumptions. Examples include index options, spread options, call on max options, put on min options, multiply strike options and power-product options. In the proof of the formula we exploit close connections between the optimal stopping problems associated with valuation of American options, obstacle problems and reflected backward stochastic differential equations.

math.PR↗

arXiv2015

Obstacle problem for semilinear parabolic equations with measure data

We consider the obstacle problem with two irregular reflecting barriers for the Cauchy-Dirichlet problem for semilinear parabolic equations with measure data. We prove the existence and uniqueness of renormalized solutions of the problem and well as results on approximation of the solutions by the penaliztion method. In the proofs we use probabilistic methods of the theory of Markov processes and the theory of backward stochastic differential equations.

math.AP↗

arXiv2014

Reflected BSDEs in time-dependent convex regions

We prove existence and uniqueness of solutions of reflected backward stochastic differential equations in time-dependent adapted and càdlàg convex regions $\mathcal{D}=\{D_t;t\in[0,T]\}$. We also show that the solution may be approximated by solutions of backward equations with reflection in appropriately defined discretizations of $\mathcal{D}$ and by a modified penalization method. The approximation results are new even in the one-dimensional case.

math.PR↗

arXiv2014

Reflected BSDEs on Filtered Probability Spaces

We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, $L^p$ data and generators satisfying weak integrability conditions. We deal with equations on general filtered probability spaces. In case the generator does not depend on the $z$ variable, we first consider the case $p=1$ and we only assume that the underlying filtration satisfies the usual conditions of right-continuity and completeness. Additional integrability properties of solutions are established if $p\in(1,2]$ and the filtration is quasi-left continuous. In case the generator depends on $z$, we assume that $p=2$, the filtration satisfies the usual conditions and additionally that it is separable. Our results apply for instance to Markov-type reflected backward equations driven by general Hunt processes.

math.PR↗

arXiv2014

Semi-Dirichlet forms, Feynman-Kac functionals and the Cauchy problem for semilinear parabolic equations

In the first part of the paper we prove various results on regularity of Feynman-Kac functionals of Hunt processes associated with time dependent semi-Dirichlet forms. In the second part we study the Cauchy problem for semilinear parabolic equations with measure data involving operators associated with time-dependent forms. Model examples are non-symmetric divergence form operators and fractional laplacians with possibly variable exponents. We first introduce a definition of a solution resembling Stampacchia's definition in the sense of duality and then, using the results of the first part, we prove the existence, uniqueness and regularity of solutions of the problem under mild assumptions on the data.

math.AP↗

arXiv2013

Dirichlet forms and semilinear elliptic equations with measure data

We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right-hand side of the equation is monotone and satisfies mild integrability assumption, and the measure is smooth. We also study regularity of solutions under the assumption that the measure is smooth and has finite total variation. Some applications of our general results are given.

math.AP↗