SubjectsSubjects(version: 861)
Course, academic year 2019/2020
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 2019 to 2019
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Guarantor: RNDr. Petr Čoupek, Ph.D.
Class: M Mgr. FPM
M Mgr. FPM > Volitelné
M Mgr. PMSE > Povinně volitelné
Classification: Mathematics > Probability and Statistics
Pre-requisite : {NMTP432 nebo NMFM408}
Annotation -
Last update: RNDr. Jitka Zichová, Dr. (24.04.2019)
In the present course stochastic linear and bilinear systems with continuous time and continuous state space are studied. The course is focused on three topics: a) optimal control b) filtering (problem of incomplete observation) 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 obtaining credit for the course are positive result in the oral exam and active participation during the exercise classes. The nature of the latter excludes the possibility of retry.

Literature -
Last update: RNDr. Petr Čoupek, Ph.D. (27.10.2019)

[1] B. Oksendal: Stochastic Differential Equations, 1st ed., Springer-Verlag, 1985.

[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.

Teaching methods -
Last update: T_KPMS (16.05.2013)


Requirements to the exam -
Last update: RNDr. Petr Čoupek, Ph.D. (27.10.2019)

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 the Riccati equation). Everything from Theorem 2.24 up to Corollary 2.38 (definitions and precise formulations of the statements). Proofs of Theorem 2.24 and Theorem 2.37.

2. Filtering: Precise statement of Kalman-Bucy filter (Theorem 3.1 without proof), application in Examples (as those discussed during the course).

3. Parameter estimation: Heuristic derivation by the least squares and maximum likelihood methods, strong consistency and asymptotic normality (Theorem 4.1 and Theorem 4.3 with proofs), statements of SLLN and CLT for martingales (Theorem 4.2 and Theorem 4.5 without proofs). To have an idea how to verify the 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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