SubjectsSubjects(version: 945)
Course, academic year 2018/2019
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Applied Stochastic Analysis - NMTP533
Title: Aplikovaná stochastická analýza
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2018
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. RNDr. Bohdan Maslowski, DrSc.
Class: M Mgr. PMSE
M Mgr. PMSE > Povinně volitelné
Classification: Mathematics > Probability and Statistics
Pre-requisite : {NMTP432 nebo NMFM408}
Is interchangeable with: NSTP240
Annotation -
Last update: RNDr. Jitka Zichová, Dr. (24.04.2019)
In the present course stochastic linear and bilinear systems with countinuous time and continuous state space are studied. The course has been focused on three topics: a) optimal control for finite and infinite time horizon control problems b) fundamentals of filtering theory c) problems of inference, parameter estimation.
Aim of the course -
Last update: T_KPMS (16.05.2013)

The goal of the course is to present some basic achievements of the stochastic control and filtering theory and related topics for linear and bilinear multidimensional systems with continuous time and continuous state space

Course completion requirements -
Last update: RNDr. Petr Čoupek, Ph.D. (04.10.2019)

The conditions for passing out in the subject are positive result in the oral exam and active participation at the exercise classes. The nature thereof excludes the possibility of repetition.

Literature - Czech
Last update: RNDr. Jitka Zichová, Dr. (24.04.2019)

[1] B. Oksendal: Stochastic Differential Equations, Springer-Verlag, 1985 (1. vyd.)

[2] W .H. Fleming and R. W .Rishel: Deterministic and Stochastic Optimal Control, Springer-Verlag, 1975

[3] J. Yong and X. Y. Zhou: Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer-Verlag, 1999

[4] P. Mandl: Pravděpodobnostní dynamické modely, Academia, Praha, 1985

Teaching methods -
Last update: prof. RNDr. Bohdan Maslowski, DrSc. (25.09.2020)

Lecture+exercises.

Requirements to the exam
Last update: RNDr. Petr Čoupek, Ph.D. (04.10.2019)

Exam Requirements

(may be slightly modified each year according to stuff talked over)

The exam is oral.

1. Control Theory. Dynamic programming method (i.e. optimal control obtained via Riccati equation). Everything from Thm 1.5 up to Thm 1.11 (definitions and precise formulations of statements). Proofs of Thms 1.5, 1.7 and 1.11.

2. Filtering. Precise statment of the main Thm 2.1 (Kalman-Bucy filter) - without proof, application in the Examples (as those discussed during the course).

3. Parameter estimation. Heuristic derivation by the least squares method, proofs of strong consistency and asymptotic normality (Thms 3.2 and 3.4 with proofs). Statements of CLT and SLLN for martingales (without proof). To have an idea how to verify conditions of these theorems in specific situations (via ergodicity).

Syllabus -
Last update: T_KPMS (16.05.2013)

1. LQ problem for linear and bilinear stochastic equations in a vector space

2. The linear filtering problem, Kalman - Bucy filter

3. Some methods of parameter estimation for linear stochastic systems, properties of estimators

Entry requirements -
Last update: prof. RNDr. Bohdan Maslowski, DrSc. (24.05.2018)

In order to enroll in this course students should possess some basic knowledge of stochastic calculus (definition and basic properties of stochastic Ito integral, Ito formula). No preliminary knowledge of stochastic differential equations is needed.

 
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