Suchergebnis: Katalogdaten im Herbstsemester 2017
Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
| Wahlfächer aus Bereichen der angewandten Mathematik ...|
Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
|Auswahl: Numerische Mathematik|
|401-4657-00L||Numerical Analysis of Stochastic Ordinary Differential Equations |
Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods"
|W||6 KP||3V + 1U||A. Jentzen|
|Kurzbeschreibung||Course on numerical approximations of stochastic ordinary differential equations driven by Wiener processes. These equations have several applications, for example in financial option valuation. This course also contains an introduction to random number generation and Monte Carlo methods for random variables.|
|Lernziel||The aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues.|
|Inhalt||Generation of random numbers|
Monte Carlo methods for the numerical integration of random variables
Stochastic processes and Brownian motion
Stochastic ordinary differential equations (SODEs)
Numerical approximations of SODEs
Multilevel Monte Carlo methods for SODEs
Applications to computational finance: Option valuation
|Skript||Lecture Notes are available in the lecture homepage (please follow the link in the Learning materials section).|
|Literatur||P. Glassermann: |
Monte Carlo Methods in Financial Engineering.
Springer-Verlag, New York, 2004.
P. E. Kloeden and E. Platen:
Numerical Solution of Stochastic Differential Equations.
Springer-Verlag, Berlin, 1992.
|Voraussetzungen / Besonderes||Prerequisites:|
Mandatory: Probability and measure theory,
basic numerical analysis and
basics of MATLAB programming.
a) mandatory courses:
Probability Theory I.
b) recommended courses:
Start of lectures: Wednesday, September 20, 2017
Date of the End-of-Semester examination: Wednesday, December 20, 2017, 13:00-15:00; students must arrive before 12:30 at ETH HG E 19.
Room for the End-of-Semester examination: ETH HG E 19.
Exam inspection: Monday, March 5, 2018,
13:00-14:00 at HG D 5.1
Please bring your legi.
|401-4785-00L||Mathematical and Computational Methods in Photonics||W||8 KP||4G||H. Ammari|
|Kurzbeschreibung||The aim of this course is to review new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods used to address challenging problems in nanophotonics. The emphasis will be on analyzing plasmon resonant nanoparticles, super-focusing & super-resolution of electromagnetic waves, photonic crystals, electromagnetic cloaking, metamaterials, and metasurfaces|
|Lernziel||The field of photonics encompasses the fundamental science of light propagation and interactions in complex structures, and its technological applications. |
The recent advances in nanoscience present great challenges for the applied and computational mathematics community. In nanophotonics, the aim is to control, manipulate, reshape, guide, and focus electromagnetic waves at nanometer length scales, beyond the resolution limit. In particular, one wants to break the resolution limit by reducing the focal spot and confine light to length scales that are significantly smaller than half the wavelength.
Interactions between the field of photonics and mathematics has led to the emergence of a multitude of new and unique solutions in which today's conventional technologies are approaching their limits in terms of speed, capacity and accuracy. Light can be used for detection and measurement in a fast, sensitive and accurate manner, and thus photonics possesses a unique potential to revolutionize healthcare. Light-based technologies can be used effectively for the very early detection of diseases, with non-invasive imaging techniques or point-of-care applications. They are also instrumental in the analysis of processes at the molecular level, giving a greater understanding of the origin of diseases, and hence allowing prevention along with new treatments. Photonic technologies also play a major role in addressing the needs of our ageing society: from pace-makers to synthetic bones, and from endoscopes to the micro-cameras used in in-vivo processes. Furthermore, photonics are also used in advanced lighting technology, and in improving energy efficiency and quality. By using photonic media to control waves across a wide band of wavelengths, we have an unprecedented ability to fabricate new materials with specific microstructures.
The main objective in this course is to report on the use of sophisticated mathematics in diffractive optics, plasmonics, super-resolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The book merges highly nontrivial multi-mathematics in order to make a breakthrough in the field of mathematical modelling, imaging, and optimal design of optical nanodevices and nanostructures capable of light enhancement, and of the focusing and guiding of light at a subwavelength scale. We demonstrate the power of layer potential techniques in solving challenging problems in photonics, when they are combined with asymptotic analysis and the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions.
In this course we shall consider both analytical and computational matters in photonics. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, spectral analysis, mathematical imaging, optimal design, stochastic modelling, and analysis of wave propagation phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in photonics, requires a deep understanding of the different scales in the wave propagation problem, an accurate mathematical modelling of the nanodevices, and fine analysis of complex wave propagation phenomena. An emphasis is put on mathematically analyzing plasmon resonant nanoparticles, diffractive optics, photonic crystals, super-resolution, and metamaterials.
|401-4645-67L||Numerics for Computational Uncertainty Quantification||W||10 KP||3V + 2U||C. Schwab|
|Kurzbeschreibung||The course presents the mathematical foundation of various numerical methods|
for the efficient quantification of uncertainty in partial differential equations.
Mathematical foundations include high dimensional polynomial approximation,
sparse grid approximations, generalized polynomial chaos expansions and their
summability properties, as well the computer implementation in model problems.
|Lernziel||The course will provide a survey of the mathematical properties and|
the computational realization of the most widely used numerical methods
for uncertainy quantification in PDEs from engineering and the sciences.
In particular, Monte-Carlo, Quasi-Monte Carlo and their multilevel extensions
for PDEs, Sparse grid and Smolyak approximations, stochastic collocation
and Galerkin discretizations will be discussed.
|Skript||There will be typed lecture notes.|
|Voraussetzungen / Besonderes||Completed BSc MATH or equivalent.|
|Auswahl: Wahrscheinlichkeitstheorie, Statistik|
|401-4607-67L||Schramm-Loewner Evolutions||W||4 KP||2V||W. Werner|
|Kurzbeschreibung||This course will be an introduction to Schramm-Loewner Evolutions which are natural random planar curve that arise in a number of contexts in probability theory and statistical theory in two dimensions.|
|Lernziel||The goal of the course is to provide an overview of the definition and the main properties of Schramm-Loewner Evolutions (SLE).|
|Inhalt||Most of the following items will be covered in the lectures: |
- Introduction to SLE
- Definition of SLE
- Phases of SLE, hitting probabilities
- How does one prove that an SLE is actually a curve?
- Restriction, locality
- Relation to loop-soups and the Gaussian Free Field
- Some SLEs as scaling limit of lattice models
|401-4597-67L||Probability on Transitive Graphs||W||4 KP||2V||V. Tassion|
|Kurzbeschreibung||In this course, we will present modern topics at the interface between probability and geometric group theory. We will define two random processes on Cayley graphs: the simple random walk and percolation, and discuss their respective behaviors depending on the geometric properties of the underlying group.|
|Lernziel||Present in an original framework important tools in the study of |
- random walks: spectral gap, harmonic functions, entropy,...
- percolation: uniqueness of the infinite cluster, mass-transport principle,...
|Inhalt||In this course, we will present modern topics at the interface between probability and geometric group theory. To every group with a finite generating set, one can associate a graph, called Cayley graph. (For example, the d-dimensional grid is a Cayley graph associated to the group Z^d.) Then, we will define two random processes on Cayley graphs: the simple random walk and percolation, and discuss their respective behaviors depending on the geometric properties of the underlying group. The focus will be on the random processes and their properties, and we will use very few notions of geometric group theory.|
|Literatur||Probability on trees and network (R. Lyons, Y. Peres)|
|Voraussetzungen / Besonderes||- Probability Theory|
- No prerequisite on group theory, all the background will be introduced in class.
|401-4619-67L||Advanced Topics in Computational Statistics||W||4 KP||2V||N. Meinshausen|
|Kurzbeschreibung||This lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling.|
|Lernziel||Students learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes.|
|Inhalt||The main focus will be on graphical models in various forms: |
Markov properties of undirected graphs; Belief propagation; Hidden Markov Models; Structure estimation and parameter estimation; inference for high-dimensional data; causal graphical models
|Voraussetzungen / Besonderes||We assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics.|
|401-4637-67L||On Hypothesis Testing||W||4 KP||2V||F. Balabdaoui|
|Kurzbeschreibung||This course is a review of the main results in decision theory.|
|Lernziel||The goal of this course is to present a review for the most fundamental results in statistical testing. This entails reviewing the Neyman-Pearson Lemma for simple hypotheses and the Karlin-Rubin Theorem for monotone likelihood ratio parametric families. The students will also encounter the important concept of p-values and their use in some multiple testing situations. Further methods for constructing tests will be also presented including likelihood ratio and chi-square tests. Some non-parametric tests will be reviewed such as the Kolmogorov goodness-of-fit test and the two sample Wilcoxon rank test. The most important theoretical results will reproved and also illustrated via different examples. Four sessions of exercises will be scheduled (the students will be handed in an exercise sheet a week before discussing solutions in class).|
|Literatur||- Statistical Inference (Casella & Berger)|
- Testing Statistical Hypotheses (Lehmann and Romano)
|401-3628-14L||Bayesian Statistics||W||4 KP||2V||F. Sigrist|
|Kurzbeschreibung||Introduction to the Bayesian approach to statistics: Decision theory, prior distributions, hierarchical Bayes models, Bayesian tests and model selection, empirical Bayes, computational methods, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods.|
|Lernziel||Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis.|
|Inhalt||Topics that we will discuss are:|
Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, Jeffreys priors), tests and model selection (Bayes factors, hyper-g priors in regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods)
|Skript||A script will be available in English.|
|Literatur||Christian Robert, The Bayesian Choice, 2nd edition, Springer 2007.|
A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013).
Additional references will be given in the course.
|Voraussetzungen / Besonderes||Familiarity with basic concepts of frequentist statistics and with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.|
|401-0625-01L||Applied Analysis of Variance and Experimental Design||W||5 KP||2V + 1U||L. Meier|
|Kurzbeschreibung||Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power.|
|Lernziel||Participants will be able to plan and analyze efficient experiments in the fields of natural sciences. They will gain practical experience by using the software R.|
|Inhalt||Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power.|
|Literatur||G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000.|
|Voraussetzungen / Besonderes||The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software R, for which an introduction will be held.|
|401-0649-00L||Applied Statistical Regression||W||5 KP||2V + 1U||M. Dettling|
|Kurzbeschreibung||This course offers a practically oriented introduction into regression modeling methods. The basic concepts and some mathematical background are included, with the emphasis lying in learning "good practice" that can be applied in every student's own projects and daily work life. A special focus will be laid in the use of the statistical software package R for regression analysis.|
|Lernziel||The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling.|
|Inhalt||The course starts with the basics of linear modeling, and then proceeds to parameter estimation, tests, confidence intervals, residual analysis, model choice, and prediction. More rarely touched but practically relevant topics that will be covered include variable transformations, multicollinearity problems and model interpretation, as well as general modeling strategies. |
The last third of the course is dedicated to an introduction to generalized linear models: this includes the generalized additive model, logistic regression for binary response variables, binomial regression for grouped data and poisson regression for count data.
|Skript||A script will be available.|
|Literatur||Faraway (2005): Linear Models with R|
Faraway (2006): Extending the Linear Model with R
Draper & Smith (1998): Applied Regression Analysis
Fox (2008): Applied Regression Analysis and GLMs
Montgomery et al. (2006): Introduction to Linear Regression Analysis
|Voraussetzungen / Besonderes||The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software package R, for which an introduction will be held.|
In the Mathematics Bachelor and Master programmes, the two course units 401-0649-00L "Applied Statistical Regression" and 401-3622-00L "Regression" are mutually exclusive. Registration for the examination of one of these two course units is only allowed if you have not registered for the examination of the other course unit.
Findet dieses Semester nicht statt.
|W||4 KP||2V||P. L. Bühlmann|
|Kurzbeschreibung||"High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed.|
|Lernziel||Knowledge of methods and basic theory for high-dimensional statistical inference|
|Inhalt||Lasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling|
|Literatur||Peter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag. |
|Voraussetzungen / Besonderes||Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics).|
|401-4623-00L||Time Series Analysis|
Findet dieses Semester nicht statt.
|W||6 KP||3G||keine Angaben|
|Kurzbeschreibung||Statistical analysis and modeling of observations in temporal order, which exhibit dependence. Stationarity, trend estimation, seasonal decomposition, autocorrelations,|
spectral and wavelet analysis, ARIMA-, GARCH- and state space models. Implementations in the software R.
|Lernziel||Understanding of the basic models and techniques used in time series analysis and their implementation in the statistical software R.|
|Inhalt||This course deals with modeling and analysis of variables which change randomly in time. Their essential feature is the dependence between successive observations.|
Applications occur in geophysics, engineering, economics and finance. Topics covered: Stationarity, trend estimation, seasonal decomposition, autocorrelations,
spectral and wavelet analysis, ARIMA-, GARCH- and state space models. The models and techniques are illustrated using the statistical software R.
|Literatur||A list of references will be distributed during the course.|
|Voraussetzungen / Besonderes||Basic knowledge in probability and statistics|
Findet dieses Semester nicht statt.
|Kurzbeschreibung||This course provides an introduction to statistical Monte Carlo methods. This includes applications of simulations in various fields (Bayesian statistics, statistical mechanics, operations research, financial mathematics), algorithms for the generation of random variables (accept-reject, importance sampling), estimating the precision, variance reduction, introduction to Markov chain Monte Carlo.|
|Lernziel||Stochastic simulation (also called Monte Carlo method) is the experimental analysis of a stochastic model by implementing it on a computer. Probabilities and expected values can be approximated by averaging simulated values, and the central limit theorem gives an estimate of the error of this approximation. The course shows examples of the many applications of stochastic simulation and explains different algorithms used for simulation. These algorithms are illustrated with the statistical software R.|
|Inhalt||Examples of simulations in different fields (computer science, statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (quantile transform, accept-reject, importance sampling), simulation of Gaussian processes and diffusions. The precision of simulations, methods for variance reduction. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC).|
|Skript||A script will be available in English.|
|Literatur||P. Glasserman, Monte Carlo Methods in Financial Engineering.|
B. D. Ripley. Stochastic Simulation. Wiley, 1987.
Ch. Robert, G. Casella. Monte Carlo Statistical Methods.
Springer 2004 (2nd edition).
|Voraussetzungen / Besonderes||Familiarity with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.|
| Auswahl: Finanz- und Versicherungsmathematik|
In den Master-Studiengängen Mathematik bzw. Angewandte Mathematik ist auch 401-3913-01L Mathematical Foundations for Finance als Wahlfach anrechenbar, aber nur unter der Bedingung, dass 401-3888-00L Introduction to Mathematical Finance nicht angerechnet wird (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (www.math.ethz.ch/studiensekretariat).
|401-4621-67L||Capita Selecta in Extreme Value Theory||W||4 KP||2V||P. Embrechts|
|Kurzbeschreibung||In this course topics beyond one-dimensional Extreme Value Theory (EVT) will be discussed. Capita Selecta included are: multivariate extremes, EVT for stationary processes, point process methodology and max-stable processes.|
|Lernziel||Students following this course will obtain an overview of modern EVT and be able to read and understand the more recent literature on the stochastic modelling of extremal events.|
|Inhalt||Topics treated will include:|
- A brief overview of one-dimensional EVT
- More-dimensional EVT
- The point process approach
- An introduction to max-stable processes
- Some applications
|Skript||This is a blackboard course typically aimed at students in mathematics.|
|Literatur||- P. Embrechts, C. Klueppelberg, T. Mikosch (1997): Modelling Extremal Events for Insurance and Finance, Springer, Berlin.|
- S.I. Resnick (1987): Extreme Values, Regular Variation, and Point Processes, Springer, New York.
- S.I. Resnick (2007): Heavy-Tail Phenomena. Probabilistic and Statistical Modeling, Springer, New York.
- Recent research papers.
|Voraussetzungen / Besonderes||The ideal background for this course is 401-3916-60 V "An Introduction to the modelling of Extremes". At the start of the present course, a brief overview of the main results from the latter course will be given. A background in measure theoretic probability theory is expected.|
|401-3925-00L||Non-Life Insurance: Mathematics and Statistics||W||8 KP||4V + 1U||M. V. Wüthrich|
|Kurzbeschreibung||The lecture aims at providing a basis in non-life insurance mathematics which forms a core subject of actuarial sciences. It discusses collective risk modeling, individual claim size modeling, approximations for compound distributions, ruin theory, premium calculation principles, tariffication with generalized linear models, credibility theory, claims reserving and solvency.|
|Lernziel||The student is familiar with the basics in non-life insurance mathematics and statistics. This includes the basic mathematical models for insurance liability modeling, pricing concepts, stochastic claims reserving models and ruin and solvency considerations.|
|Inhalt||The following topics are treated:|
Collective Risk Modeling
Individual Claim Size Modeling
Approximations for Compound Distributions
Ruin Theory in Discrete Time
Premium Calculation Principles
Tariffication and Generalized Linear Models
Bayesian Models and Credibility Theory
|Skript||M. V. Wüthrich, Non-Life Insurance: Mathematics & Statistics|
|Voraussetzungen / Besonderes||The exams ONLY take place during the official ETH examination period.|
This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch.
Prerequisites: knowledge of probability theory, statistics and applied stochastic processes.
|401-3922-00L||Life Insurance Mathematics||W||4 KP||2V||M. Koller|
|Kurzbeschreibung||The classical life insurance model is presented together with the important insurance types (insurance on one and two lives, term and endowment insurance and disability). Besides that the most important terms such as mathematical reserves are introduced and calculated. The profit and loss account and the balance sheet of a life insurance company is explained and illustrated.|
|401-3928-00L||Reinsurance Analytics||W||4 KP||2V||P. Antal, P. Arbenz|
|Kurzbeschreibung||History of reinsurance and catastrophic events. Forms of reinsurance. Modelling of reinsurance losses through frequency severity models. Rating/Pricing of reinsurance contracts. Modelling of natural catastrophes. Reinsurance markets and companies. Risk profile and solvency implications of reinsurance. Solvency 2 modelling. Alternatives to reinsurance such as Cat Bonds.|
|Lernziel||Understand the following aspects: History of reinsurance. Role of reinsurance in society and history of catastrophic events. Forms of reinsurance (proportional and nonproportional). Covered types of business (property, casualty, specialties, life, health). Modelling of reinsurance losses through frequency severity models (typical distributions and parameters). Rating/Pricing of reinsurance contracts (experience and exposure). Modelling of natural catastrophes (methodological approaches and techniques). Natural catastrophes in Switzerland (importance, insurance, reinsurance). Reinsurance markets and companies. Risk profile implications of reinsurance (Catastrophe risk, reserving risk, Credit risk, basis risk, etc). Solvency implications of reinsurance (primary insurance and reinsurance side). Solvency 2 modelling (standard models, internal models, FINMA StandRe). Alternatives to reinsurance (insurance linked securities, subordinate debt). Trigger types of cat bonds (indemnity, modeled loss, industry loss, parametric)|
|Inhalt||History of reinsurance. Role of reinsurance in society and history of catastrophic events. Forms of reinsurance (proportional and nonproportional). Covered types of business (property, casualty, specialties, life, health). Modelling of reinsurance losses through frequency severity models (typical distributions and parameters). Rating/Pricing of reinsurance contracts (experience and exposure). Modelling of natural catastrophes (methodological approaches and techniques). Natural catastrophes in Switzerland (importance, insurance, reinsurance). Reinsurance markets and companies. Risk profile implications of reinsurance (Catastrophe risk, reserving risk, Credit risk, basis risk, etc). Solvency implications of reinsurance (primary insurance and reinsurance side). Solvency 2 modelling (standard models, internal models, FINMA StandRe). Alternatives to reinsurance (insurance linked securities, subordinate debt). Trigger types of cat bonds (indemnity, modeled loss, industry loss, parametric)|
|Skript||Slides, lecture notes, and references to literature will be made available.|
|401-3927-00L||Mathematical Modelling in Life Insurance||W||4 KP||2V||T. J. Peter|
|Kurzbeschreibung||The course covers various mathematical models that are used in life insurance.|
|Lernziel||The course's objective is to present various mathematical models that are used in life insurance for valuation or risk management purposes.|
|Inhalt||Following main topics are covered:|
1. Guarantees & options in life insurance
2. Financial modeling
3. Valuation of life insurance contracts: Unit linked and participating contracts
4. Mortality modeling
|Voraussetzungen / Besonderes||The exams ONLY take place during the official ETH examination period.|
The course counts towards the diploma of "Aktuar SAV".
Basic knowledge in probability theory and stochastic processes is assumed. Some knowledge in financial mathematics is useful.
|401-4935-67L||Mean Field Games||W||4 KP||2V||M. Burzoni|
|Kurzbeschreibung||The analysis of differential games with a large number of players finds applications in various research fields, from physics to economics and finance. The aim of Mean Field Games theory is to provide a suitable approximation of such problems with a higher tractability.|
|Lernziel||This course aims to give a broad understanding of the basic ideas of Mean Field Games, the main mathematical tools and the possible applications.|
|Inhalt||We first present and analyze toy models of Mean Field Games in order to familiarize with the subject and to understand what kind of problems can be solved with this theory.|
We recall some basic principles of optimal control theory and stochastic differential equations.
We explore two different approaches to Mean Field Games. From an analytic point of view it consists of a coupled system of PDEs. From a probabilistic point of view it amounts to a particular type of stochastic differential equations. We will concentrate, in particular, in the probabilistic approach.
|Literatur||1) Notes on Mean Field Games. P. Cardaliaguet|
2) Mean Field Games. J.M. Lasry, P.L. Lions
3) Probabilistic theory of Mean Field Games and applications. R. Carmona, F. Delarue
|Voraussetzungen / Besonderes||Basic courses in analysis including basic knowledge of ordinary/partial differential equations.|
Basic knowledge of stochastic analysis including Brownian Motion and stochastic differential equations.
|401-4923-67L||Polynomial Jump-Diffusions and Applications in Finance||W||4 KP||2V||M. Larsson|
|Kurzbeschreibung||A basic goal in mathematical finance is to develop market models that combine statistical flexibility with analytical tractability. A common class of such models are affine, and more generally polynomial, jump-diffusions. This course will develop the theory of polynomial jump-diffusions, the mathematical tools needed to study them, and discuss a selection of applications.|
|Lernziel||The aim of this course is to develop the theory of polynomial jump-diffusions, the mathematical tools needed to study them, and discuss a selection of applications.|
|Inhalt||- Introduction to affine and polynomial processes|
- Semimartingales and their characteristics; jump-diffusions
- Affine and polynomial jump-diffusions; the moment formula; the exponential-affine transform formula
- Existence and uniqueness theory: Martingale problems; the positive maximum principle; SDE methods
- Invariance properties
- Applications: Optimal investment; term structure of interest rates; credit risk
- Advanced topics: Volterra processes with affine characteristics
|Literatur||The course is based on class notes. References for additional and background reading will be provided on the course website.|
|Voraussetzungen / Besonderes||Basic knowledge of stochastic analysis including Brownian Motion and Stochastic Calculus.|
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