Zakład Teorii Prawdopodobieństwa i Matematyki Finansowej

Kierownik:

• prof. dr hab. Łukasz Stettner (prof. zw.)
  email
  pok. 318, tel.: 22 5228 126

Pracownicy:

• prof. dr hab. Krzysztof Bogdan (prof. zw.)
  email
• prof. dr hab. Tomasz Komorowski (prof. zw.)
  email
  pok.115, tel.: 22 5228 115
• prof. dr hab. Szymon Peszat (prof. nadzw.)
  email
• dr Paweł Wolff (adiunkt)
  email
  pok.508, tel.: 22 5228 176
• prof. dr hab. Jerzy Zabczyk (prof. zw.)
  email
  pok. 309a, tel.: 22 5228 184
• mgr Piotr Żebrowski (asystent)
  email

Doktoranci:

• Michał Krzemiński
  email
• Michał Krzeszowiec
  email
• Tomasz Rogala
  email

O Zakładzie

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. Brzeźniak). 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. Brzeźniak) 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.

Ł. 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]  Ł. 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 Ł. Stettner, Nearly Optimal Controls for Stochastic Ergodic Problems with Partial Observation, SIAM J. Control Optimiz. 31 (1993), 180 - 218.

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

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

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

[11]  Ł. 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:

• 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 semi-classical sets. I-III, Bull. Acad. Polon. Sci. 12 (1964), 265 - 270; 13 (1965), 147 - 150, 215 - 219.
• A construction of basis in C¹(I²), 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. Brzeźniak), Space-time 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 Navier-Stokes 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 one-dimensional diffusions, Bernoulli 11 (2005), 247 - 262.
• (with J. Zabczyk), Stochastic Partial Differential Equations with Lévy Noise, Cambridge Univ. Press, 2007.

Ł. 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. Pasik-Duncan), 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 discrete-time Markov processes with long run average cost, Systems Control Lett. 34 (1998), 55 - 62.
• (with D. Gątarek), 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 discrete-time parameter systems in Hilbert spaces, SIAM J. Control Optim. 13 (1975), 1217 - 1234.
• A note on C₀-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. Gołdys), Ornstein-Uhlenbeck semigroups in open sets of Hilbert spaces, C. R. Acad. Sci. Paris Série I 325 (1997), 433 - 438.
• (with A. Karczewska), Stochastic PDEs with function-valued solutions, Proceedings of the Colloquium ”Infinite-Dimensional 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 non-local operators, Journal on Functional Analysis 216 (2004), 455 - 490.
• (with S. Peszat), Stochastic Partial Differential Equations with Lévy Noise, Cambridge Univ. Press, 2007.

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