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2 Bulletin of the Polish Academy of Sciences. Mathematics

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

Klimsiak T., Rozkosz A.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

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

Klimsiak T., Rozkosz A.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

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.