There are no prerequisites.

# Modelling with stochastic processes

Actuarial part:

Lundberg process, the probablity of ruin, martingale methods, the probablity of ruin in finite time, generalized Lundberg model.

Markov chain models, Kolmogorov equations,

Thiele differential equation, mathematical reserves calculation, reserves dependent payoffs, stochastic interest rates via Markov chains.

Financial part:

Optimal control: formulation of the problem,

Hamilton-Jacobi-Bellman equations, linear regulator, applications.

Optimal stopping: formulation of the problem,

examples, American options.

Fundamental theorem of asset pricing: formulation, proof, hedging equations, connections to partial differential equations, examples of incomplete markets.

Incomplete markets: Lévy models, superhedging, pricing, optimization.

- T. Björk: Arbitrage theory in continuous time, 2nd ed., Oxford : Oxford University Press, 2004.
- H. Bühlmann: Mathematical methods in risk theory, Berlin : Springer, cop. 1996, 2005.
- M. Denuit ... [et al.]: Actuarial theory for dependent risks : measures, orders and models, Chichester : J. Wiley & Sons, cop. 2005.
- I. Karatzas, S. E. Shreve: Methods of mathematical finance, 2nd ed., New York : Springer, cop. 1998.
- B. Øksendal: Stochastic differential equations : an introduction with applications, 6th ed., Berlin : Springer, cop. 2005.
- W. Schoutens: Lévy processes in finance : pricing financial derivatives, Chichester (West Sussex) : Wiley, cop. 2003.
- D. Wong: Generalised optimal stopping problems and financial markets, Harlow (England) : Longman, cop. 1996.

Stochastic processes form a basis for numerous models in finance and insurance. The course links theoretical parts learned in other courses on stochastic processes by showing their applications on one side and elucidates

the theoretical background on the other.

Since the content is of great practical importance we expect that also specialists from financial practice will present their work experience during the course.

Knowledge and understanding: Understanding of stochastic modelling in finance and insurance and understanding of mathematical framework.

Application: Application is immediate as the models under consideration form a basis for

Pricing many financial and insurance products.

Reflection: The application of stochastic processes deepens the knowledge of probability calculus and stochastic processes and paves the way for their application.

Transferable skills: The skills obtained are transferable to other areas of mathematical modelling, but the gist of the course is its immediate applicability.

Lectures, exercises, homeworks, consultations, seminars

Seminar work

Oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Janez Bernik:

BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Realizing irreducible semigroups and real algebras of compact operators. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2008, vol. 348, no. 2, str. 692-707. [COBISS-SI-ID 14899289]

BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Positivity and matrix semigroups. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2011, vol. 434, iss. 3, str. 801-812. [COBISS-SI-ID 15745625]

BERNIK, Janez, MARCOUX, Laurent W., RADJAVI, Heydar. Spectral conditions and band reducibility of operators. Journal of the London Mathematical Society, ISSN 0024-6107, 2012, vol. 86, no. 1, str. 214-234. [COBISS-SI-ID 16357721]