Department of Probability Theory and Mathematics of Finance



About the Department

Main research results of the members of the Department are described below and a list of papers, selected by the authors, is presented. Complete lists of publications can be found through links.
The Department seminar

Z. Ciesielski

In probability: Simple construction of Brownian motion in terms of Schauder bases. Determining the exact Hausdorff measure of the Brownian trajectories (with S. J. Taylor). Discovering the principle of not feeling the boundary in heat conduction. The Wiener measure is concentrated on a suitable Hölder-Orlicz class with exponent ½. Application of Schauder spline bases to calculating the fractal dimension of realizations of random fields.

In approximation theory: Characterization of Hölder classes by the coefficients of Schauder and Franklin expansions. Description of the basic properties of the Franklin orthogonal system; exponential estimates. Positive solution to the Banach problem on existence of a basis in the space of continuously differentiable functions on the square. Building up (jointly with J. Domsta and T. Figiel) a theory of spline bases on compact sets whose idea preceded the wavelet theory.

S. Peszat

In infinite dimensional stochastic analysis: Establishing the Freidlin-Ventsel large deviation estimates for a general class of diffusions in Hilbert spaces. Proving (jointly with J. Zabczyk) that the transition semigroup corresponding to a stochastic evolution equation is strong Feller and irreducible, provided that the nonlinearities are Lipschitz continuous. Consequently, the invariant measure for infinite dimensional diffusion is unique. Establishing sufficient conditions for existence and properties of solutions of infinite dimensional stochastic equations with Lipschitz nonlinearities (jointly with Zabczyk) and polynomial nonlinearities (jointly with Z. Brze¼niak). Formulating (with J. Zabczyk) necessary and sufficient conditions for the existence of function-valued solutions to multidimensional stochastic wave and heat equations.

Establishing (jointly with M. Capiński and Z. Brze¼niak) the existence and uniqueness of solutions to stochastic Navier-Stokes and Euler equations. Proving (with T. Komorowski) the uniqueness in law of the solution to the passive tracer problem in an irregular velocity field. Establishing (jointly with J. Zabczyk) basis for SPDEs with Lévy noise.

£. Stettner

In stochastic control:
existence of solutions to the Bellman equations corresponding the problems: partially observed control problem with average cost per unit time criterion ([3]), risk sensitive control problem with infinite time ergodic cost criterion with complete and partial observations ([5], [6], [9]),
construction of nearly optimal strategies with applications to adaptive control ([2], [4]);
in filtering theory:
conditions for ergodicity of filtering processes ([1], [8], [10]);
in mathematics of finance:
existence of optimal strategies for general utility maximization in discrete time ([7]),
existence of optimal strategies for growth optimal and risk sensitive growth optimal portfolios with transaction costs ([11]),
Related papers:

[1]  £. Stettner, On Invariant Measures of Filtering Processes, Proc. 4th Bad Honnef Conf. on Stochastic Differential Systems, Ed. N. Christopeit, K. Helmes, M. Kohlmann, Lect. Notes in Control Inf. Sci. 126, Springer 1989, 279 - 292.

[2]  W. J. Runggaldier and £. Stettner, Nearly Optimal Controls for Stochastic Ergodic Problems with Partial Observation, SIAM J. Control Optimiz. 31 (1993), 180 - 218.

[3]  £. Stettner, Ergodic Control of Partially Observed Markov Processes with Equivalent Transition Probabilities, Applicationes Mathematicae 22.1 (1993), 25 - 38.

[4]  T. Duncan, B. Pasik-Duncan, £. Stettner, Discretized Maximum Likelihood and Almost Optimal Adaptive Control of Ergodic Markov Models, SIAM J. Control Optimiz. 36 (1998), 422 - 446.

[5]  G. B. Di Masi, £. Stettner, Risk sensitive control of discrete time Markov processes with infinite horizon, SIAM J. Control Optimiz. 38 (2000), 61 - 78.

[6]  G. B. Di Masi, £. Stettner, Risk sensitive control of discrete time partially observed Markov processes with infinite horizon, Stochastics and Stochastics Rep. 67 (1999), 309 - 322.

[7]  M. Rasonyi, £. Stettner, On utility maximization in discrete - time market models, Annals of Applied Prob. 15 (2005), 1367 - 1395.

[8]  G. Di Masi, £. Stettner, Ergodicity of Hidden Markov Models, Math. Control Signals Systems 17 (2005), 269 - 296.

[9]  G. B. Di Masi, £. Stettner, Infinite horizon risk sensitive control of discrete time Markov processes under minorization property, SIAM J. Control Optimiz. 46 (2007), 231 - 252.

[10]  G. Di Masi, £. Stettner, Ergodicity of filtering process by vanishing discount approach, Systems and Control Letters 57 (2008), 150 - 157.

[11]  £. Stettner, Discrete Time Infinite Horizon Risk Sensitive Portfolio Selection with Proportional Transaction Costs, Banach Center Publications, to appear.

J. Zabczyk

In stochastic processes: Polar sets do not coincide with null sets for Lévy processes. Strong Feller property is equivalent to null controllability for linear stochastic evolution equations (with G. Da Prato). Stochastic factorization and continuity of stochastic convolution (with G. Da Prato and S. Kwapień). Smoothing properties of transition semigroups in Hilbert spaces (with G. Da Prato). Characterization of linear stochastic systems with function valued solutions (with A. Karczewska). Existence of solutions to non-linear heat and wave equations with spatially homogeneous noise (with S. Peszat). Extending Liouville theorem for non-local operators (with E. Priola).

In deterministic control: Location of spectrum does not determine the growth of a linear system. Detectability implies uniqueness of algebraic Riccati equation in infinite dimensions. Analysis of spectral properties of null-controllable systems with vanishing energy (with E. Priola).

In stochastic control: Analysis of algebraic Riccati equation of discrete time stochastic infinite-dimensional systems. Continuous time version of the best choice (with R. Cowan).

Selected publications

Z. Ciesielski:

S. Peszat:

£. Stettner:

J. Zabczyk: