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Ważniejsze wydarzenia w teorii procesów stochastycznych

Zabczyk J.

Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne

2004

[Z] 40

77-95

Bellman's inclusions and excessive measures

Zabczyk J.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

101--122

The paper is concerned with Bellman's inclusions for the value function of the optimal stopping for a Markov process X on a complete separable metric space E. The author investigates a connection between seemingly unrelated objects; excessive measures, differential inclusions and optimal stopping. Conditions are given under which an evolutionary Bellman inclusion has a strong or weak solution in the Hilbert space L2 (E, µ), where p is an excessive measure for X. The solution is identified with the value function of a stopping problem. The stationary Bellman inclusion is treated as well. Specific examples of diffusions with jumps and infinite-dimensional diffusions are discussed. Excessivity of the measure µ plays an essential role in the development. The results are then applied to pricing American options both in finite and infinite dimensions recently investigated by Zhang [32], Mastroeni and Matzeu [20], [21], and Gątarek and Musiela [11].

Invariant measures for stochastic heat equations

Tessitore G., Zabczyk J.

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 2

271--287

The paper is concerned with the asymptotic behaviour of solutions to the nonlinear stochastic heat equations, with spatially homogeneous noise, in the whole space. Sufficient conditions for the existence of invariant measures, in weighted spaces of locally square-integrable functions, are given. For linear equations with multiplicative noise an invariant measure, supported by positive functions, is constructed. The existence of a stationary solution to the vector Burgers equations is obtained as an application of the general theory.