Actuarial and Financial Sciences

ALESSANDRO DE GREGORIO

General goals
The main goal of the course is to introduce advanced random processes and probabilistic tools that are particularly useful in quantitative finance and actuarial sciences.

Knowledge and understanding
By the end of the course, students will be able to understand the meaning of random patterns (e.g., with jumps) arising in the study of financial and insurance topics.

Applying knowledge and understanding
Students will acquire the skills necessary to model complex phenomena through the theoretical concepts explored in depth during the lectures. In particular, the advanced stochastic analysis tools studied during the course will enable students to address some actuarial and financial issues.

Making judgements
By the end of the course, students will be able to critically analyze phenomena that evolve randomly over time and are subject to random shocks. Furthermore, students will develop the sensitivity necessary to choose models best suited to the study of such complex systems.

Communication skills
Students will develop communication skills useful for describing random phenomena through the language of mathematics and probability. These skills will emerge through understanding the intuitive aspects related to the mathematical tools underlying stochastic processes.

Learning skills
Students during the course will study stochastic concepts and methods that will enable them to understand subsequent courses in finance and actuarial sciences.

The student will be able to use the fundamental advanced tools for understanding random models used in finance and actuarial sciences (including jump processes). In addition, students will become familiar with certain methods based on partial differential equations (for example, the Feynman-Kac formula).

Knowledge of the basic concepts of Probability and Stochastic Calculus is required. Furthermore, the student must be familiar with the tools of calculus.

1. Stochastic processes. General facts. Kolmogorov's existence theorem. Construction of a random process.
2. Conditional mean: definition. Properties and convergence results. Stopping times.
3, Martingale theory. Definitions and examples. Doob-Meyer decomposition theorem. Optional stopping theorem.
4. Brownian motion. Construction of a Brownian motion. Markov processes. Properties of the sample paths: continuity and non-differentiability. Quadratic variation.
5. Stochastic integral. Definition of Ito's integral. Properties and applications. Stochastic calculus. Ito's lemma and its applications. Girsanov theorem and Cameron-Martin formula.
6. Stochastic differential equations. Definition and examples. Existence and uniqueness. Functional of Feynman-Kac. Applications: finance, actuarial sciences, epidemiology.
7. Hints on the simulation of stochastic differential equations.

R. Cont, P. Tankov, Financial modeling with jump processes, Chapman & Hall 2004
S.E. Shreve, Stochastic Calculus for Finance II, Springer 2004
J.M. Steele, Stochastic Calculus and Financial Applications, Springer 2000

Karatzas, I., Shreve S.E. (1998) Brownian Motion and Stochastic Calculus. Springer.
Oksendal, B. (2010) Stochastic Differential Equations: An Introduction with Applications. Springer.

Frontal teaching.

Attendance is not compulsory, however it is strongly recommended.

The written and oral test tends to evaluate students' exhibition skills and understanding of the basic concepts discussed during the lectures.

1) Definition of Brownian Motion.

2) Ito integral and its definition.

3) Examples of Stochastic Differential Equations.