Department of Probability Theory and Mathematics of Finance
Head:
Staff:
 Tomasz Byczkowski (Professor)
email
Division in Wroc³aw
 Tomasz Komorowski (Professor)
email
pok.115, tel.: 22 5228 115
 Szymon Peszat (Professor)
email
Division in Kraków
 Pawe³ Wolff (Assistant Professor)
email
pok.508, tel.: 22 5228 176
 Jerzy Zabczyk (Professor)
email
pok. 309a, tel.: 22 5228 184
Phd Students:
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 HölderOrlicz 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 FreidlinVentsel 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 functionvalued solutions
to multidimensional stochastic wave and heat equations.
Establishing (jointly with M. Capiński and Z. Brze¼niak) the existence
and uniqueness of solutions to stochastic NavierStokes 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. PasikDuncan, £. 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 nonlinear heat and wave
equations with spatially homogeneous noise (with S. Peszat). Extending
Liouville theorem for nonlocal 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
nullcontrollable systems with vanishing energy (with E. Priola).
In stochastic control: Analysis of algebraic Riccati equation of discrete
time stochastic infinitedimensional systems. Continuous time version of the
best choice (with R. Cowan).
Selected publications
Z. Ciesielski:

On the isomorphisms of the spaces H_{α} and m,
Bull. Acad. Polon. Sci. 8 (1960), 217  222.
 (with S. J. Taylor), First passage times and sojourn times
for the Brownian motion in
the space and the exact Hausdorff measure of the sample path,
Trans. Amer. Math. Soc. 103 (1962),
434  450.

Properties of the orthonormal Franklin system. I, II,
Studia Math. 23 (1963), 141  157; 27 (1966), 87  121.

Brownian motion, capacitory potentials and semiclassical
sets. IIII,
Bull. Acad. Polon. Sci. 12 (1964), 265  270; 13 (1965), 147  150, 215  219.

A construction of basis in C^{1}(I^{2}),
Studia Math. 33
(1969), 243  247.

Constructive function theory and spline systems, Studia
Math. 53 (1975), 277  302.
 (with T. Figiel), Spline bases in classical function spaces
on compact C^{∞}
manifolds. I, II, Studia Math. 76 (1983), 1  58, 95  136.

Asymptotic nonparametric spline density estimation,
Probab. Math. Statist. 12 (1991), 1  24.

Orlicz spaces, spline systems and Brownian motion,
Constr. Approx. 9 (1993), 191  208.

Fractal functions and Schauder bases, Comput. Math.
Appl. 30 (1995), 283  291.
S. Peszat:

Large deviation principle for stochastic evolution equations,
Probab. Theory Related Fields 98 (1994), 113  136.
 (with J. Zabczyk), Strong Feller property and
irreducibility for diffusions on Hilbert spaces,
Ann. Probab. 23 (1995), 157  172.
 Existence and uniqueness of the solution for stochastic
equations on Banach spaces,
Stochastics Stochastics Rep. 55 (1995), 167  193.
 (with J. Zabczyk), Stochastic evolution equations with a
spatially homogeneous Wiener process,
Stochastic Process. Appl. 72 (1997), 187  204.
 (with Z. Brze¼niak), Spacetime continuous solutions to SPDEs driven by a homogeneous
Wiener process, Studia Math. 137 (1999), 261  299.
 (with J. Zabczyk), Nonlinear stochastic wave and heat
equations, Probab. Theory
Related Fields 116 (2000), 421  443.
 (with M. Capiński), On the existence of a solution to
stochastic NavierStokes equations, Nonlinear Anal. 44 (2001), 141  177.
 The Cauchy problem for a nonlinear stochastic wave
equation in any dimension, J. Evol. Equ. 2 (2002), 383  394.
 (with T. Komorowski), Transport of a passive tracer by an
irregular velocity field, J. Statist. Phys. 115 (2004), 1383  1410.
 (with F. Russo), Large noise asymptotics for onedimensional
diffusions, Bernoulli 11 (2005), 247  262.
 (with J. Zabczyk), Stochastic Partial Differential Equations with
Lévy Noise, Cambridge Univ. Press, 2007.
£. Stettner:

Ergodic control of partially observed Markov processes with equivalent
transition probabilities, Appl. Math. 22 (1993), 25  38.
 (with G. B. Di Masi), Risk sensitive control of discrete time
partially observed
Markov processes with infinite horizon, in:
Proc. 37th IEEE CDC, Tampa, 1998, 3467  3472.
 (with W. Runggaldier), Approximations of Discrete Time
Partially Observed Control Problems, Appl. Math. Monographs
CNR, Giardini Ed., Pisa, 1994.
 (with T. Duncan and B. PasikDuncan), Discretized maximum
likelihood and almost optimal adaptive control of ergodic Markov models,
SIAM J. Control Optim. 36 (1998), 422  446.
 (with G. B. Di Masi), Bayesian ergodic adaptive control of
discrete time Markov
processes, Stochastics Stochastics Rep. 54 (1995), 301  316.
 (with G. B. Di Masi), Bayesian adaptive control of discretetime Markov
processes with long run average cost, Systems Control
Lett. 34 (1998), 55  62.
 (with D. G±tarek), On the compactness method in general
ergodic impulsive
control of Markov processes, Stochastics Stochastics Rep. 31
(1990), 15  26.
 (with G. B. Di Masi), Risk sensitive control of discrete
time Markov processes
with infinite horizon, SIAM J. Control Optim. 38 (1999), 61  78.
J. Zabczyk:
 On optimal stochastic control of discretetime parameter
systems in Hilbert spaces,
SIAM J. Control Optim. 13 (1975), 1217  1234.
 A note on C_{0}semigroups, Bull. Acad. Polon. Sci. 23
(1975), 895  898.
 Remarks on algebraic Riccati equation in Hilbert spaces,
J. Appl. Math. Optim. 2 (1976), 251  258.
 (with R. Cowan), An optimal selection problem associated
with the Poisson problem,
Theory Probab. Appl. 23 (1978), 606  614.
 (with G. Da Prato and S. Kwapień), Regularity of solutions
of linear stochastic equations in
Hilbert spaces, Stochastics Stochastics Rep. 23 (1987), 1  23.
 (with G. Da Prato), Smoothing properties of the Kolmogoroff semigroups in Hilbert spaces,
Stochastics Stochastics Reports 35 (1991), 63  77.
 Mathematical Control Theory. An Introduction, Birkhäuser, 1992.

Chance and Decision. Stochastic Control in Discrete Time,
Quaderni Scuola Norm. Sup. Pisa, 1992.
 (with G. Da Prato), Stochastic Equations in Infinite Dimensions,
Cambridge Univ. Press, 1992.
 (with G. Da Prato), Regular densities of invariant measures
in Hilbert spaces, J. Funct. Anal. 130 (1995), 427  449.
 (with S. Peszat), Strong Feller property and irreducibility
for diffusions on Hilbert spaces,
Ann. Probab. 23 (1995), 157  172.
 (with G. Da Prato), Ergodicity for Infinite Dimensional Systems,
Cambridge Univ. Press, 1996.
 (with S. Peszat), Stochastic evolution equations with a
spatially homogeneous Wiener process,
Stochastic Process. Appl. 72 (1997), 187  204.
 (with G. Da Prato, B. Go³dys),
OrnsteinUhlenbeck semigroups in open sets of Hilbert spaces,
C. R. Acad. Sci. Paris Série I 325 (1997), 433  438.
 (with A. Karczewska),
Stochastic PDEs with functionvalued solutions,
Proceedings of the Colloquium ”InfiniteDimensional Stochastic Analysis”
of the Royal Netherlands Academy of Arts and Sciences, Amsterdam 1999, Eds. Ph. Clement, F. den Hollander,
J. van Neerven and B. de Pagter, North Holland, 197  216.
 (with S. Peszat),
Nonlinear stochastic wave and heat equations,
PTRF 116 (2000), 421  443.
 (with G. Da Prato), Second Order
Partial Differential Equations in Hilbert Spaces, Cambridge Univ. Press,
2002.
 (with E. Priola),
Null controllability with vanishing energy,
SIAM Journal on Control and Optimization 42 (2003), 1013  1032.
 (with E. Priola),
Liouville theorems for nonlocal operators,
Journal on Functional Analysis 216 (2004), 455  490.
 (with S. Peszat), Stochastic Partial Differential
Equations with Lévy Noise, Cambridge Univ. Press, 2007.