COURSE OUTLINE
Stochastic Analysis
1. General
School
Academic Unit
Level of Studies
Course code
Semester
Course Title
Idependent Teaching Activities
Weekly Teaching Hours
Credits
Course Type
Prerequite Courses
Language of Instruction and Examinations
Is the course offered to Erasmus Students?
Url (Eclass)
2. Learning Outcomes
Learning Outcomes
Students are given the opportunity to deepen their knowledge into well-known concepts from Probability Theory, and to understand and familiarize to new notions and theoretical constructions such as e.g. those of the conditional expectations with respect to an information domain (σ-algebra), the martingales and the Brown motion, the Itô integral, and others. The aim of the course is the understanding of the basic concepts of Stochastic Analysis, in such a way that students will be able to implement them in actuarial applications, insurance and finance.
Upon successful completion of the course, students will be able to:
- realise and be able use the general construction of the measure
- realise the concept of a σ-algebra and to be able to perform calculations upon conditional to certain σ-fields
- understand the random behaviour as determined by a martingale process, its properties and to be able to distinguish such type of behaviours in insurance, finance and other fields of application (e.g. economy, environment, etc)
- implement the Brownian motion in modelling stochastic phenomena understanding the benefits and the limits of such approximations
- understand the different approximation perspective offered by Ito integration, its properties and the central role that plays in stochastic finance and insurance through standard models
General Competences
- Working independently
- Teamwork
- Production of new research ideas
- Production of free, creative, and inductive thinking
3. Syllabus
- Elements of measure-theoretic probability: information structures (σ-algebras), random variables and the information structures they generate, measurability of random variables, and induced probability measures.
- Probability spaces and integration: probability measure spaces, integration with respect to a probability measure, convergence theorems, conditional expectation and its main properties, conditioning on events, random variables, and σ-algebras.
- Martingales and Brownian motion: martingale processes and filtrations; basic properties and examples of martingales in discrete time, including simple random walks and Markov processes; stopping times; martingale convergence results and inequalities; Brownian motion, its basic properties, increments and sample paths; simulation schemes.
- Itô processes and stochastic differential models: Itô processes, the Itô integral and its properties, Itô’s lemma for diffusion processes, stochastic modelling of standard models arising in finance and actuarial science, and numerical simulation schemes.
4. Teaching and Learning Methods - Evaluation
Delivery
Use of Information and Communications Technology
- Distance learning by using the asynchronous platform e-class.
- Distance learning by using the synchronous platform MS Teams.
- Projectors
- Communication via email
Teaching Methods
Activity
Semester Workload
Student Performance Evaluation
Language of evaluation: Greek
Methods of evaluation: Final exam and exercises
The final grade (G) is determined from the grade of exercises (E) and the grade in final exam (F) from the formula:
G = max {F, 0.30*E + 0.70*F}.
The final exam consists of multiple-choice questionnaires, short-answer questions and problem solving.
For ERASMUS+ students it is possible to have a take home exam or a project/assignment, in English, as a method of evaluating their performance.
5. Attached Bibliography
Suggested Bibliography
Suggested Bibliography (in Greek)
- Βασιλείου, Π.-Χ., Γ. (2001). Στοχαστικά Χρηματοοικονομικά, Εκδόσεις Ζήτη (Κωδικός Εύδοξου: 11280)
- Γιαννακόπουλος, Α. Ν. (2003). Στοχαστική Ανάλυση και Εφαρμογές στην Χρηματοοικονομική, Πανεπιστημιακές Σημειώσεις (Διαθέσιμες online)
- Χελιώτης, Δ. (2015). Εισαγωγή στο Στοχαστικό Λογισμό, Πανεπιστημιακές Σημειώσεις (Διαθέσιμες online)
Suggested Bibliography (in English)
- Brzezniak, Z., Zastawniak, T. (2000). Basic stochastic processes: a course through exercises. Springer Science & Business Media.
- Capiński, M., & Kopp, P. E. (2004). Measure, integral and probability (Vol. 14). Springer.
- Choe, G. H. (2016). Stochastic analysis for finance with simulations, Springer.
- Karatzas, I., Shreve, S. (2014). Brownian motion and stochastic calculus, Springer.
- Klebaner, F. C. (2005). Introduction to stochastic calculus with applications, Imperial College Press.
- Shreve, S. E. (2004). Stochastic calculus for finance II: Continuous-time models (Vol. 11). Springer.