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A note on Dirichlet processes

100%

Rozkosz A.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

tom Vol. 48, no 3

277-286

We consider an m-symmetric conservative Hunt process (Chi, Rho[chi]) on Chi such that for some phi [belongs to] L2(Chi; my) the composite process phi(Chi) is for quasi every starting point chi [belongs to] Chi a continuous Rho[chi]-Dirichlet process in the sense of Foellmer. We show that the martingale and the zero quadratic variation parts in the decomposition od phi(Chi} can be chosen to be additive functionals and moreover, how to pass from Foellmer's decomposition to Fukushima's and the Lyons-Zheng decompositions of phi(Chi).

Probabilistic solutions of quasilinear PDE's

100%

Rozkosz A.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

tom Vol. 48, no 1

103-111

We prove existence of probabilistic solutions of quasilinear Cauchy problem for second order operator under mild regularity assumptions on its coefficients. In the proof we use Krylov-Safonov estimates and Krylov's estimates on stochastic integrals.

Time-inhomogeneous diffusions corresponding to symmetric divergence form operators

100%

Rozkosz A.

Probability and Mathematical Statistics

2002

tom Vol. 22, Fasc. 2

231--252

We consider a time-inhomogeneous Markov family (X, P3,x) corresponding to a symmetric uniformly elliptic divergence form operator. We show that for any φ in the Sobolev space W1p∩W12 with p = 2 if d = 1 and p > d > if d > 1 the additive functional Xφ = (φ (X1)- φ (Xx); 0 ≤ s < t} admits a unique strict decomposition into a martingale additive functional of finite energy and a continuous additive functional of zero energy. Moreover, we give a stochastic representation of the zero energy part and show that in case the diffusion coefficient is regular in time the functional Xφ is a Dirichlet process for each starting point (s, x). The paper contains also rectifications of incorrectly presented or incorrectly proved statements of our earlier paper [14].

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

63%

Klimsiak T. , Rozkosz A.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

tom Vol. 65, no. 1

45--56

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.

On Backward Stochastic Differential Equations Approach to Valuation of American Options

63%

Klimsiak T. , Rozkosz A.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

tom Vol. 59, no 3

275--288

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.