Course syllabus for Stochastic calculus

The course syllabus contains changes
See changes

Course syllabus adopted 2021-02-26 by Head of Programme (or corresponding).

Overview

  • Swedish nameStokastisk analys
  • CodeTMS165
  • Credits7.5 Credits
  • OwnerMPENM
  • Education cycleSecond-cycle
  • Main field of studyMathematics
  • DepartmentMATHEMATICAL SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

  • Teaching language English
  • Application code 20130
  • Open for exchange studentsYes

Credit distribution

0104 Examination 7.5 c
Grading: TH		7.5 c
  • 25 Okt 2022 am J
  • 04 Jan 2023 pm J
  • 18 Aug 2023 am J

In programmes

Examiner

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Eligibility

General entry requirements for Master's level (second cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

English 6 (or by other approved means with the equivalent proficiency level)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

Probability theory from a first course in mathematical statistics.

Some experience of computer programming such as, for example, basic knowledge of Matlab programming.

Mathematics corresponding to what students from quite math oriented educations such as TM (Technical Mathematics) or F (Engineering Physics) learn on basic level.

Aim

Give good understanding of and ability to use those aspects of stochastic calculus and stochastic differential equations that are of the greatest importance in both applications in technical sciences and natural sciences as well as in further mathematical and mathematical statistical theory development.

Learning outcomes (after completion of the course the student should be able to)

Describe defining properties for stochastic differential equations and their solutions together with ability to use basic methods solution methods thereof including quadratic variation, martingale techniques and Ito's formula. Describe the relation between solutions to stochastic differential equations and solutions to certain (deterministic) partial differential equations from a principal theoretical point of view as well as performing corresponding computational examples. Describe change of measure and change of drift coefficient for solutions to stochastic differential equations from a principal theoretical point of view as well as by application to statistical inference. Describe basic principles for numerical solution of stochastic differential equations according to the course literature from a principal theoretical point of view as well as performing corresponding computational examples.

Content

Variation and quadratic variation of functions. Review of Riemann integral, Riemann-Stieltjes integral and Lebesgue integral. Introduction to axiomatic probability theory and to abstract conditional expectation with respect to sigma fields. Brownian motion (Wiener process) together with its most important properties. Defining properties of martingales and Markov processes with continuous time and continuous values. Ito integrals, Ito integral processes and Ito¿s formula. Stochastic differential equations together with existence, uniqueness and Markov property of weak solutions and strong solutions thereof. Stochastic exponential, stochastic logarithm and linear stochastic differential equations.  Stratonovich stochastic calculus. Kolmogorov¿s equations together with Dynkin¿s formula and the Feynman-Kac formula. Time homogeneous diffusion processes together with explosion, recurrence, transience and stationary distributions thereof. Change of probability measure for stochastic variables. Change of probability measure and drift coefficient for solutions of stochastic differential equations with applications to likelihood principles and statistical inference.

Organisation

Lectures and tutorials.

Literature

Chapters 1-6 and 10 in "Fima C. Klebaner (2012). Introduction to Stochastic Calculus with Applications, Third Edition, Imperial College Press, London". Additional lecture notes on applications and numerical methods.

Examination including compulsory elements

Written exam.

The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers on educational support due to disability.

The course syllabus contains changes

  • Changes to examination:
    • 2023-03-20: Exam date Exam date 2023-08-18 added by Elisabeth Eriksson
      [35921, 58593, 3], New exam for academic_year 2022/2023, ordinal 3 (not discontinued course)
    • 2022-08-25: Exam date Exam date changed by Elisabeth Eriksson
      [35921, 58593, 2], New exam for academic_year 2022/2023, ordinal 2 (not discontinued course)