Stochastic processes in natural and social phenomena
Course Status: Elective
Office hours: upon agreement
Very few processes are completely deterministic – an element of randomness is almost always present. The adequate mathematical framework to treat this randomness is the theory of stochastic processes. They are abundant in practically all aspects of nature and society. The theory of stochastic processes is part of the mathematical theory of probability, however, in this course the pragmatic approach will be taken that the basic laws will be introduced to the students through the analysis of natural and social phenomena. Relatively limited mathematical prerequisites will be sufficient for attending the course (elementary calculus and some basic knowledge about differential equations). After a short overview of the fundamentals of probability theory we will discuss random processes from physics, population dynamics, epidemiology and finance. We will discuss the basic stochastic processes and the tools to handle them always in a heuristic, application-oriented manner.
Week no. Session title
What are stochastic processes? Their abundance. Examples from physics, population dynamics, epidemics and finance.
2. Probability theory
Random variables, distributions. Independent variables. Conditional probability. Averages and moment. Central limit theorem. Dependencies. Multivariate problems.
3. Poisson process
Radioactivity, random queuing problem, birth-and-death processes. Waiting time distribution. Properties of the Poisson process.
4. Renewal processes
Problems of insurance policy. Generalization of the Poisson process to arbitrary waiting time distributions. Waiting time paradox. Renewal equation.
5. Markov processes
The Markov property. Brownian motion and random walk. Discrete random walk. Further examples (population dynamics, economics).
Markov chains, transition matrix, representation as directed graphs.
6. Branching processes
Examples: Genealogy, cascading. The tree topology. The problem of extinction. Tree-approximation to networks with loops.
Master (rate) equation
Ergodicity. Continuous time Markov processes. Transition probabilities from empirical data. Stationary solution and relaxation to it.
7. Fokker-Planck (forward Kolmogorov) equation
Diffusion and migration in space. Derivation of the FP equation, interpretation of the terms. Simple solutions.
8. Continuous time random walks
Generalization of the Poisson process. Montroll-Weiss equation. Fat tailed distributions. Applications to financial time series.
9. First passage time problems
Expected lifetimes of patients, the Gambler’s ruin problem. The problem of stability in a noisy environment. Relation to risk estimation. First passage time of the one-dimensional Brownian motion. Characteristic times in the limit order book.
10. Extreme value statistics
The definition of the problem. Applications: reliability and financial risk estimation. The three basic cases and distributions. Consequences for stochastic processes.
11. Stochastic differential equations
The Langevin equation of the Brownian particle. Types of noise, white noise. The Wiener and the Ornstein Uhlenbeck processes. Geometric Brownian motion and the Black-Scholes equation of option pricing.
12. Non-Markovian processes
Colored noise, memory effects. Power spectrum, 1/fα noise. Water level fluctuations of the Nile, Hurst exponent. Fractional Brownian motion. White to colored noise transformation. ARCH-GARCH models of return fluctuations.
Small world model
Watts-Strogatz model: rewiring and adding shortcuts. Calculation of the shortest path, degree distribution and clustering coefficient. Scaling.
Power law degree distribution
Barabási-Albert model. Temporal evolution. Degree distribution. Clustering coefficient. Small and ultrasmall worlds. Vertex copying models. Geographical models.
Motifs and their calculation. Communities. Basic detection methods. Benchmarks.
Weighted network models
Different sources of weights. Min cut max flow theorem. Intensity and coherence. Weighted motifs and communities.
Diffusion and spreading on networks
The diffusion problem. Diffusion on the ER, WS, BA graphs. Stationary solutions, relaxation. Searching on networks.
Basic epidemic spreading models. Threshold phenomena. Mean field solutions. Epidemic models on graphs. The dynamics of spreading.
Bursty signals and temporal networks
Burstiness as basic feature of human behavior. Communication networks are temporal. Characterization. Temporal motifs.
A. Barrat, M. Barthélemy and A. Vespignani: Dynamic Processes on Networks (Cambridge, 2008)
N.G. van Kampen: Stochastic Processes (North Holland, 1992)
M.E.J. Newman: Networks. An introduction (Oxford UP, 2010)
W. Paul and J. Baschnagel: Stochastic Processes – From Physics to Finance (Springer, 1999)
Further literature will be suggested during the course.
- Students will learn, what stochastic processes are and will get acquainted with a broad set of examples from the fields of physics, population dynamics, epidemiology and finance;
- They will learn the basics of the theory of stochastic processes;
- They will be able to recognize the need of applying tools from the theory of stochastic processes in solving real world problems;
- They will acquire skills to solve simple problems in the field of stochastic processes.
(1) Assessment 1 (20% of final grade): Students home works. Simple exercises to deepen understanding.
(2) Assessment 2 (35% of final grade): Midterm test.
(3) Assessment 3 (45% of final grade): Final test. Answering questions to account for understanding the basic concepts.
Elements of calculus, differential equations.