Stochastic operations research
Finite and infinite geometric series summation, the power series of the exponential function, solution of linear systems, computation of expectations for functions of random variables, through integration and summation, the exponential function, the Poisson process, discrete time Markov chains, renewal processes, continuous time Markov chains, introduction to queuing theory (M/M/1, M/M/s, finite population models, birth and death queues, M/G/1, G/M/1, G/G/1, G/G/s), open and closed networks of queues. All topics will be handled both through the use of a computer and without the help of technology.


    Financial mathematics
    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.


    Pre-master statistics
    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.


    Pre-master stochastic operations research
    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.