Non-Gaussian stochastic processes in Nature with elements of econo- and sociophysics

The student should be familiar with basic elements of probability theory and mathematical statistics. In particular,

(i) with the most significant elements of limit theorems such as central limit theorem and Bernoulli law of large numbers;

(ii) Benoulli distribution and its relation with Gauss and Poisson distributions;

(iii) Winer process (or Brownian motion)

(i) fundamental elements of Markov chain and process.

Remote learning
Self-reading

The lectures have an interdisciplinary character and are meant for the graduate students who are interested in hot topics of advanced statistical physics and chaotic dynamics which are connected with non-Gaussian stochastic processes and some of their applications, for instance, in econo- and sociophysics.

The lectures have an interdisciplinary character and is directed to the graduate students who are interested in hot topics of advanced statistical physics and chaotic dynamics which are connected with non-Gaussian stochastic processes and some of their applications. During the lectures is presented their experimental basis; mainly non-Debye relaxation of photocurrents in amorphous films. The non-Gaussian (mainly Lévy processes) are considered by using the continuous-time random walk formalism and Weierstrass-Mandelbrot reprersentation of the basic, so called waiting-time distribution. Concluding: the role of the non-Gassian processes in physics and in financial markets is emphasized.

Contents:

I The Gaussian processes and introduction to the non-gaussian ones

1. Brownain motion, critical opalescence and sky-blue

2. The role of fluctuations - approaches of Einstein and Smoluchowski

3. Markow stochastic process, Fick's law and central limit theorem (CLT)

4. Characteristic function and cumulants

5. Determination of the Avogadro number: Perrin experiment

6. Geometric Brownian motion and Blacka-Scholes pricing of options: econophysical

interpretation of Fick's diffusion

7. CLT and power-laws

8. Polimer diffusion - long-term correlations: breaking of CLT

9. Financial time-series and moving average

10. Autocorrelaftion function and spectral density

11. Volatility and higher-order correlations

II Non-Gaussian and non-Markowian stochastic processes - extended CLT

12. Mathematical fractals and physical fractals (prefraktals)

13. Dispersive transport and long-time relaxation

14. Continuous-Time Random Walk

15. Renormalization group techniques

16. Hierarchical trapping in deterministic chaos - Weierstrass-Mandelbrot flights and walks

17. Lévy processes: stochastic hierarchicity, self-similarity & self-affinity, inhomogeneous

scaling relation, singularity and extreme events

18. Generalized master equatin (with memory), fractional Fokker-Planck equation, telegraphic equation, anomalous diffusion

Prerequisites:

Statystka dla fizyków, Fizyka statystyczna, Mechanika statystyczna

Examination: Examination

Basic literature

1 J. Haus and K. W. Kehr, "Diffusion in Regular and Disordered Lattices", Physics Reports 150 (1987) 263

2 J.-P. Bouchaud and A. Georges, "Anomalous Diffusion in Disordered Media: Statistical Mechanisms, Models and Physical Applications", Phys. Rep. 195 (1990) 127

3 L.P. Kadanoff, "From Order to Chaos. Essays: Critical, Chaotic and Otherwise", World Scient. Series on Nonlinear Science Series A, Vol.1, ser. Ed. L.O. Chua (World Scient., Singapore 1993)

4 M. F. Schlesinger, G. M. Zaslavsky, U. Frisch (Eds.) "Levy Flights and Related Topics in Physics", Lecture Notes in Physics 450 (Springer-Verlag, Berlin 1995)

5 A. Bunde and S. Havlin (Eds.) "Fractals in Science" (Springer-Verlag, Berlin 1995)

6 A. Bunde and S. Havlin (Eds.) "Fractals in Disordered Systems" (Second Revised and Enlarged Edition, Springer-Verlag, Berlin 1996)

7 R. Kutner, A. Pękalski, K. Sznajd-Weron (Eds.) "Anomalous Diffusion. From Basis to Applications", Lecture Notes in Physics, 519 (Springer-Verlag, Berlin 1999)

8 W. Paul and J. Baschnagel, "Stochastic Processes. From Physics to Finance" (Springer-Verlag, Berlin (1999)

9 R. N. Mantegna and H. E. Stanley, "An Introduction to Econophysics. Correlations and Complexity in Finance" (Cambridge Univ. Press, Cambridge 2000; tłumaczenie PWN 2001)

10 D. Sornette, "Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools" (Springer-Verlag, Berlin 2000)

11 J.-P. Bouchaud and M. Potters, "Theory of Financial Risks. From Statistical Physics to Risk Management" (Cambridge University Press, Cambridge 2001)

12 J. Czekaj, M. Woś, J. Żarnowski, "Efektywnośc giełdowego rynku akcji w Polsce" (PWN, Warszawa (2001).

Additional literature

J. Klafter, M. F. Shlesinger, G. Zumofen, "Beyond Brownian Motion", Physics Today 49 (1996) 33

M. Zaslavsky, "Chaotic dynamics and the origin of statistical laws", Physics Today, 52 (1999) 39

D. Stauffer and H.E. Stanley, "From Newton to Mandelbrot. A primar in theoretical physics with fractals for the personal computer" (Springer-Verlag, Berlin 1996).

S. Chandrasekhar, M. Kac, R. Smoluchowski, "Marian Smoluchowski His Life and Scientific Work", Polish Scientific Publishers PWN, Warszawa 2000.

S. Chandrasekhar, "Stochastic Problems in Physics and Astronomy", Review of Modern Physics 15 (1943) 1

N.G. van Kampen, "Procesy stochastyczne w fizyce i chemii", PWN, Warszawa 1990.

After completion of the course the student obtained the following results in the field of education.

KNOWLEDGE

1) He knows the most important issues of the non-Gaussian stochastic chains and processes in particular knowledge on the Levy flight and walk, fat-tailed distributions and their applications in widely-understood physics.

2) He knows the basic problems of theory of extreme events.

3) He knows the canonical version of the continuous-time random walk theory.

SKILLS

1) He can distinguish Gaussian stochastic processes from the non-Gaussian ones by understanding the conditions in which both classes of processes are created and the properties that characterize them.

2) Knows how to solve problems associated with various non-Gaussian chains and stochastic processes. In particular, he knows how to read the characteristics of the processes from the empirical data.

3) Knows how to bring out the Gaussian process from the empirical time series.

ATTITUDES

1) Appreciates the importance of a thorough and comprehensive understanding of the problem in drawing conclusions and making decisions.

It all corresponds to the following effects in the field of education (see information about studying at http://www.fuw.edu.pl/ ):

1) knowledge: KW01- KW06,

2) skills: KU05 - KU09,

3) competences: K04, K05.

EXPECTED STUDENT WORKLOAD:

- participation in classes (lectures 30h + exercises 30h): 60h - 2.0 ECTS,

- preparation for classes and the dissolution of homework: 60h - 2.0 ECTS,

- exam preparation: : 40h - 1.5 ECTS.

1) A necessary condition for taking the exam is to pass the exercise.

2) The exam is oral in nature and can be performed in three methods:

- the seminar method, where student chooses three subjects, preparing presentations and refers two of them chosen by the teacher,

- using the project agreed with the lecturer,

- using the traditional exam.

It is not expected