Lecture: Mi, 14-16h, HS II, Albertstr. 23b, Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: David Criens
Assistant: Samuel Adeosun
Language: in English
This lecture builds the foundation of one of the key areas of probability theory: stochastic analysis. We start with a rigorous construction of the It^o integral that integrates against a Brownian motion (or, more generally, a continuous local martingale). In this connection, we learn about It^o's celebrated formula, Girsanov’s theorem, representation theorems for continuous local martingales and about the exciting theory of local times. Then, we discuss the relation of Brownian motion and Dirichlet problems. In the final part of the lecture, we study stochastic differential equations, which provide a rich class of stochastic models that are of interest in many areas of applied probability theory, such as mathematical finance, physics or biology. We discuss the main existence and uniqueness results, the connection to the martingale problem of Stroock-Varadhan and the important Yamada-Watanabe theory.
Probability Theory I and II (Stochastic Processes)
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Applied Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Advanced Lecture in Stochastics (MScData24)
Elective in Data (MScData24)
Undergraduate seminar: Counter-Examples in Probability Theory
Teacher: David Criens
Assistant: Andreas Demleitner
It is not only theorems, proofs or illustrative examples, but also counterexamples that show the depth and beauty of a theory. Natural questions are: (a) are the requirements of a theorem necessary and not only sufficient; (b) are the requirements sufficient and not just necessary; (c) is an implication an equivalence, i.e. does the implication in the other direction also hold.
In this undergraduate seminar we deal with counter-examples from probability theory. Possible topics range from classic questions such as measurability, independence of random variables, expectations or conditional probabilities, to more advanced topics such as limit value rates, martingals or Markov processes. A suitable topic can be found for any interested student.
Elementary Probability Theory I (topics from probability theory I–III can also be assigned)
Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)
Lecture: Di, Mi, 16-18h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
24.02., 14:00-16:00, HS II, Albertstr. 23b
Teacher: David Criens
Assistant: Eric Trébuchon
Language: in German
Complex analysis deals with functions \(f : \mathbb C \to \mathbb C\) , which map complex numbers to complex numbers. Many concepts of Analysis~I can be directly transferred to this case, e.\,g. the definition of differentiability. One might expect that this would lead to a theory analogous to Analysis~I but much more is true: in many respects you get a more elegant and simpler theory. For example, complex differentiability on an open set implies that a function is even infinitely often differentiable, and this is further consistent with analyticity. For real functions, all these notions are different. However, some new ideas are also necessary: For real numbers \(a\), \(b\) one integrates for \[\int_a^b f(x) \mathrm dx\] over the elements of the interval \([a, b]\) or \([b, a]\). However, if \(a\), \(b\) are complex numbers, it is no longer so clear clear how such an integral is to be calculated. One could, for example, in the complex numbers along the line that connects \(a, b \in \mathbb C\), or along another curve that leads from \(a\) to \(b\). Does this lead to a well-defined integral term or does such a curve integral depend on the choice of the curve?
Required: Analysis I+II, Linear Algebra I
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Mathematical Concentration (MEd18, MEH21)
Pure Mathematics (MSc14)
Elective (MSc14)
Elective (MScData24)
Lecture: Do, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
Teacher: David Criens
Assistant: Dario Kieffer
Language: in English
The class of Markov chains is an important class of (discrete-time) stochastic processes that are used frequently to model for example the spread of infections, queuing systems or switches of economic scenarios. Their main characteristic is the Markov property, which roughly means that the future depends on the past only through the current state. In this lecture we provide the mathematical foundation of the theory of Markov chains. In particular, we learn about path properties, such as recurrence and transience, state classifications and discuss convergence to the equilibrium. We also study extensions to continuous time. On the way we discuss applications to biology, queuing systems and resource management. If the time allows, we also take a look at Markov chains with random transition probabilities, so-called random walks in random environment, which is a prominent model in the field of random media.
Required: Elementary Probability Theory~I \ Recommended: Analysis~III, Probability Theory~I
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Applied Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective in Data (MScData24)
Mo, 10-12h, HS 00-036, Georges-Köhler-Allee 101
Teacher: David Criens
Assistant: Timo Enger
Mi, 16-17h, HS II, Albertstr. 23b
Teacher: David Criens, Peter Pfaffelhuber, Angelika Rohde, Thorsten Schmidt
Lecture: Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Teacher: David Criens
general:
Mi, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Teacher: Angelika Rohde, David Criens
Supplementary Module in Mathematics (MEd18)
Compulsory Elective in Mathematics (BSc21)
Mi, 16-17h, SR 226, Hermann-Herder-Str. 10
Teacher: David Criens, Angelika Rohde, Thorsten Schmidt
Lecture: Fr, 10-12h, HS Weismann-Haus, Albertstr. 21a
31.07., 15:30-18:00
Sit-in exam (resit) 13.02., 10:00-12:00
Teacher: David Criens
Assistant: Timo Enger
general: ,
Compulsory Elective in Mathematics (BSc21)
Fr, 13-14h, HS II, Albertstr. 23b
Teacher: David Criens, Peter Pfaffelhuber, Angelika Rohde, Thorsten Schmidt
Lecture: Fr, 10-12h, HS Weismann-Haus, Albertstr. 21a
23.02., 13:00-16:00
Teacher: David Criens
Assistant: Timo Enger
general: ,
Elementary Probability Theory I (BSc21, MEB21, MEdual24)
Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Teacher: David Criens
Assistant: Lars Niemann
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Fr, 13-14h, HS II, Albertstr. 23b
Teacher: David Criens, Peter Pfaffelhuber
Teacher: David Criens
Assistant: Saskia Glaffig