Stella Kapodistria

This is an introductory course in stochastic operations research. As such, you cannot avoid being introduced to several basic ideas and principles that can prove quite challenging at first. However, regardless whether you plan to go to industry after finishing your degree or if you are planning on further studies, you will certainly use material from this class.

Learning objectives:
The objective of the course is to help the students master the basic tools used in analysing systems that operate in the presence of randomness. At the end of the course, students should be able to:

1. ◘ Model real life/practical problems using Markov chains (MC) and renewal processes (RP) – Modelling
2. ◘ Use the theory of Markov chains, renewal processes, and queueing theory to solve practical problems – Solving
3. ◘ Calculate the performance measures of a system through the study of MC and RP – Calculating Toolbox outcomes: solve linear systems, calculate integrals and summations, master computations of expectations, use power and geometric series
4. ◘ Define appropriate cost structures and efficiently optimize the behaviour of the system– Optimizing
5. ◘ Use mathematical software (Mathematica) for large scale numerical calculations – Coding

Short promotional description of the course:
We focus on model-based process analysis using Operation Research. Students can formulate discrete and continuous time Markov chains and queuing models for several practical situations. For discrete-time chains, we compute transient and limiting distributions, costs and first passage times. For continuous-time chains, we compute the limiting distribution, long-term costs and first passage times. Students can calculate and interpret several performance measures in a number of queuing models.

The course is designed to help you master this material. To this purpose, the course material is divided into theory, exercises/instructions and pc instructions. During all sessions, you are strongly advised to bring your laptop. It will help with solving the exercises.

Content:
Topics taught in 2DI60 include discrete and continuous time Markov chains, Poisson process, renewal processes, introduction to queueing systems (M/M/1, M/M/s, finite population models, birth and death queues, M/G/1, G/M/1, G/G/1), and open and closed networks of queues (Jackson networks). More generally, this course deals with the modelling and analysis of systems that operate in the presence of randomness/uncertainty. The goal of the course is to set the mathematical foundations so as the students understand, analyse, and ultimately optimize the behaviour of such systems.
Syllabus

This is an introductory course in financial math; we start by introducing the concept of pricing of financial derivatives by the so-called no-arbitrage arguments. We first explain the fundamental underlying economic ideas by considering the basic example of the pricing of a forward contract. Then, we introduce the concept of tactical asset allocation of capital over the various asset classes (stock, bonds, commodities, real estate, currency). We discuss the classical ideas about the trade-off between risk and return, and the famous model of the Nobel laureate Markowitz. After that, we consider the pricing problem in more detail for two fundamental models in stochastic finance: the binomial tree model and the Black-Scholes model. Among other things, we derive the celebrated Black-Scholes formula for the price of European call and put options. In addition to the theory we consider practical methods for solving financial math problems using the freely available statistical computer package R.
Syllabus

Descriptive statistics and graphical representation of data, Random variables (continuous and dicrete), Probability and its properties, Probability distributions, Probability density function, distribution function, expectation, variance, Sampling, Central Limit Theorem, Estimation (point and interval estimators), Hypothesis testing, Hypothesis testing and confidence intervals for one sample, Hypothesis testing and confidence intervals for population fractions, Simple linear regression.

Markov chains: Definition, transient behavior, limiting behavior, cost models, cohort models
Markov processes: Definition, limiting behavior, Poisson processes, birth and death processes
Renewal theory: Definition of renewal process, renewal-reward theorem with applications
Queueing models: Exponential models (M/M/1, M/M/s, M/M/s/K), Limiting distribution, performance measures like throughput, average number of customers and average waiting time
Non-exponential models (M/G/1, G/M/1, G/G/1, G/G/s), Mean value analysis for M/G/1 model, Limiting distribution for G/M/1 model, Approximations for G/G/1 and G/G/s model
Networks of queues, Open (Jackson) network with single- and multi-server stations, Closed network with single-server station.

Introduction to probability, random variables. Binomial, Poisson-distribution. Normal, exponential distribution. Mean and variance of a random variable. Central Limit Theorem, linear combination of random variables. Descriptive statistics. Estimation theory (unbiasedness and Mean Square Error), confidence intervals, principles of hypothesis testing.

The following subjects are treated:
Discrete time Markov chains, including classification of states and long run behaviour and branching processes.
Exponential distribution and Poisson Processes.
Generating functions and Laplace-Stieltjes transforms.
Continuous time Markov chains and birth-and-death processes.
Renewal theory, including renewal theorem, renewal reward processes and regenerative processes.

In deze minor staan (wiskundige) basismodellen uit de Kwantitatieve Financiering en Actuariaat centraal. Op deze manier krijgt de student een goed beeld van wat deze specialisaties inhouden. De belangrijkste thema’s zijn de beheersing van risico’s en de waardering van onzekere inkomensstromen.