Mathematik - WiSe 2024/25

This is a continuation of the lecture course Functional Analysis from the previous semester. Attendance of the latter is not a necessary prerequisite for this course. However, a background in the theory of Banach and Hilbert spaces will be required. Main topics to be covered are a proof of the spectral theorem for bounded self-adjoint operators, an introduction to unbounded symmetric and self-adjoint operators and the basics of the Fourier transform and distributions.

Einschreibeschlüssel: selbstadjungiert

Wir werden typische Aufgabenstellungen beim Staatsexamen in Analysis behandeln,
Lösungsmethoden besprechen und evtl. noch etwas zugrunde liegende Theorie
wiederholen. Wir beginnen mit dem Themenbereich
Gewöhnliche Differentialgleichungen und
arbeiten uns dann zur  Funktionentheorie  durch.

Mehrdimensionale Analysis -- Lehramt Gymnasium im Wintersemester 2024/2025
Dienstag 10-12 Uhr in B138 und Freitag 10-12 Uhr in B138

Beginn: Dienstag 15.10.2024
Die Vorlesung ist die dritte eines dreisemestrigen Kurses in Mathematik
für das Lehramt an Gymnasien nach der neuen Prüfungsordnung.

Einschreibung mit Schlüssel: ma

The lecture provides an introduction to stochastic calculus with an emphasis on the mathematical concepts that are later used in the mathematical modeling of financial markets.

In the first part of the lecture course the theory of stochastic integration with respect to Brownian motion and Ito processes is developed. Important results such as Girsanov's theorem and the martingale representation theorem are also covered. The first part concludes with a chapter on the existence and uniqueness of strong and weak solutions of stochastic differential equations.

The second part of the lecture course gives an introduction to the arbitrage theory of financial markets in continuous time driven by Brownian motion. Key concepts are the absence of arbitrage, market completeness, and the risk neutral pricing and hedging of contingent claims. Particular attention will be given to the the Black-Scholes model and the famous Black-Scholes formula for pricing call and put options.

If you wish to participate in the course, please sign up by sending an e-mail from your LMU e-mail address to Annika Steibel (steibel@math.lmu.de).

About this Lecture

This lecture discusses the interplay of Theory, Modelling, Numerical Methods and Implementation in Mathematical Finance.
All aspects learn from each other: one needs to understand the theory to build models and good implementations. Studying numerical experiments gives deeper theoretical insight. Using the computer to understand math can be fun!
We discuss how to build an industry-grade implementation of our models and allow future extensions while being efficient.
We discuss practical applications in the financial industry.

The lecture tries to be as self-contained as possible, but we will use some numerical methods developed in a previous course. We will start with a short recapitulation of the numerical methods needed for those who did not follow the last lecture. It is possible to consider most of these parts as “given” (“black box”). Don’t panic: We will assist you.

The lecture will discuss the theory and application of some prominent methods and models from mathematical finance. We focus on interest rate and hybrid models with high relevance for the financial industry. In an excursion, we consider a climate model (DICE), extend it and combine it with our interest rate models.
We will then use our implication to gain a deeper understanding of the theoretical properties of the model.

If time permits, we conclude the lecture by discussing running our models in a cloud.

Tentative Agenda

• Recapitulation of Numerical Methods (Monte-Carlo Method, SDEs, etc.)
• Interest Rates, Linear Interest Rates Products
• Multiple Interes Rate Curve Modelling and its Implementation
• Interest Rate Options, Convexity Adjustment
• Discrete Term-Structure Models (formerly known as LIBOR Market Models) and their Implementation
  • Model Calibration
  • Valuation of Complex Derivatives
  • Object Oriented Implementation, Definition of Model Interfaces
• Short Rate Models and their Implementation
• Climate Models
• Numerical Methods in the Context of Mathematical Finance
  • Model Calibration: Optimization and Root-Finding
  • Correlation Modelling: Prinzipal Component Analysis and Factor Reduction, Regression Methods
• IT Implementation of Models, e.g. in the Cloud

The lecture covers the object oriented implementation of the algorithms and using modern software development tools.
As part of the implementation of the models and the valuation algorithms, the lecture will discuss some of the latest standards in software development.

• revision control systems (git)
• unit-testing (junit)
• build management (maven, gradle)
• continuous integration (TravisCI, Jenkins)

Implementation will be performed in Java (Eclipse, IntelliJ)

Course Description

In this seminar, we will analyse one-dimensional stochastic differential equations (SDEs) driven by a Brownian motion. To this end, a short review of semimartingales, stochastic integrals and Itô's formula will be discussed. Under Lipschitz conditions on the drift and diffusion coefficients of the SDE, we will derive unique strong solutions. In particular, we will recover the Brownian bridge, the geometric Brownian motion and the Ornstein-Uhlenbeck process as solutions of affine SDEs.

Target Participants

Einschreibeschlüssel: AnIS2425

Einschreibeschlüssel: LA1WS24

Einschreibeschlüssel: Grundlagen1W24

Blockkurs in der vorlesungsfreien Zeit, in dem die Grundlagen des Textsatzsystems LaTeX und speziell die Erstellung mathematischer Texte behandelt werden. Nähere Informationen zu Ablauf und Anmeldung auf der Webseite des Kurses.

Inhalt der Vorlesung ist eine Einführung in die Optimierung in - vornehmlich - endlicher Dimension. Wichtige Themen und Inhalte sind unter anderem:

Konvexität, Lineare Programme und ihre Standardformen, Existenz von Lösungen für lineare Programme, Dualitätstheorie für lineare Programme, das Simplexverfahren, Formulierung und Existenz von Lösungen konvexer Optimierungsprobleme, duale Darstellung konvexer Funktionen und die Kuhn-Tucker-Theorie.