arXiv subjects

Tomasz Klimsiak

Publications and source records attributed to Tomasz Klimsiak.

59 records · Page 4Linked to original sources

arXiv2013

Semilinear elliptic systems with measure data

We study the Dirichlet problem for systems of the form -Δu^k=f^k(x,u)+μ^k, x\inΩ, k=1,...,n, where Ω\subset R^d$ is an open (possibly nonregular) bounded set, μ^1,...,μ^n are bounded diffuse measures on Ω, f=(f^1,...,f^n) satisfies some mild integrability condition and the so-called angle condition. Using the methods of probabilistic Dirichlet forms theory we show that the system has a unique solution in the generalized Sobolev space i.e. space of functions having fine gradient. We provide also a stochastic representation of the solution.

math.AP↗

arXiv2013

On backward stochastic differential equations approach to valuation of American options

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 weak solution to some semilinear partial differential equation.

math.PR↗

arXiv2011

On time-dependent functionals of diffusions corresponding to divergence form operators

We consider additive functionals as a time and space-dependent function of a diffusion corresponding to nonhomogeneous uniformly elliptic divergence form operator. We show that if the function belongs to natural domain of strong solutions of PDEs then there is a version of this function such that additive functional is a continuous Dirichlet process for almost every starting points of diffusion and we describe the martingale and the zero-quadratic variation parts of its decomposition. We show also that under slightly stronger condition on the function this property holds for every starting point. Finally, we prove an extension of the Itô formula.

math.PR↗

arXiv2010

Strong solutions of semilinear parabolic equations with measure data and generalized backward stochastic differential equations

We prove that under natural assumptions on the data strong solutions in Sobolev spaces of semilinear parabolic equations in divergence form involving measure on the right-hand side may be represented by solutions of some generalized backward stochastic differential equations. As an application we provide stochastic representation of strong solutions of the obstacle problem be means of solutions of some reflected backward stochastic differential equations. To prove the latter result we use a stochastic homographic approximation for solutions of the reflected backward equation. The approximation may be viewed as a stochastic analogue of the homographic approximation for solutions to the obstacle problem.

math.PR↗

arXiv2009

Reflected BSDEs and continuous solutions of parabolic obstacle problem for semilinear PDEs in divergence form

We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the eqution and mild conditions on the obstacle a unique continuous solution of the problem admits a stochastic representation in terms of reflected backward stochastic differential equations. We derive also some regularity properties of so- lutions and prove useful approximation results.

math.PR↗