17266 - Stochastic Processes

Learning outcomes

The course deals with the basics of stochastic processes with applications in insurance and finance. At the end of the course the student is able to work with some of the most important kinds of random processes such as Markov chains, Poisson processes, birth and death processes and Brownian motion.

Course contents

1. Review of basic probability concepts.
   
   Probability spaces; Random Variables; Expectations; Indipendence; Conditional distributions and conditional expectations; Porperties of the Expctation and Variance operators (marginal and contitional); Moment generating functions; Some distributions: Bernouilli, Poisson, Exponential, Gamma, Compound Poisson.
   
   
2. Discrete time Markov Chains.
   
   Transition matrix and state distribution. Homogeneous Markov Chains. Chapman-Kolmogorov equations. State classification: persistent and transient states. First passing probabilities and first passing times. Mean recurrence (return) time. Null-persistent states. Periodic states. Examples. Communicating and intercommunicating states. Close sets and irreducible sets of states. Decomposition theorem. Ergodicity and stationary distributions. Finite states Markov Chains. Non negative matrices, stochastic matrices and regular matrices. Eigendecomposition of transition matrices. Perron-Frobenious theorem.
   
   
3. Poisson and Compound Poisson processes and applications.
   
   The ruin problem: an introduction in discrete time: claim number process, total claim process, reserves and the ruin problem.
   
   A continuous time model for the total claim process (Cramèr-Lundberg process). Arrival and waiting times. Claims distribution. Premium and initial reserve. The "ruin" event, finite horizon and infinite horizon. The running maximum process.
   
   Independent and/or stationary increment processes.
   
   The homogeneous Poisson process. Five equivalent definitions. Intensity arrival times, waiting times, number of events and Poisson distribution. Simulation of trajectories. Superposition property.
   
   Compound Poisson process and Compount Poisson process with trend.
   
   The Ruin problem. Evento rovina, tempo di rovina e probabilità di rovina. Net Profit Condition. Lundberg adjustment coefficient. Lundberg inequality. The special case of Gamma (and Exponential) distributed Claims.
   
   Pricing of bonds with default risk. The non-homogeneous Poisson process.

Readings/Bibliography

Probability and Random Processes, G.R. Grimmett and D.R. Stirzaker, 4th Edition, Oxford university Press, 2020

Non-Life Insurance Mathematics, an Introduction with Stochastic Processes, T. Milkosch. Springer-Verlag, 2004.

Teaching methods

Classes, exercises and examples by means of computer simulations.

Assessment methods

A two-hours written examination that includes theoretical questions and exercises

Teaching tools

* Slides and lecture notes

* Solved exercises

* computer programs for the simulation of the presented random processes 

Office hours

See the website of Paolo Foschi