MATH43720: Stochastic Analysis

• Analysis and Probability

• None

• None

• To introduce key concepts in conditional expectations, martingale and stochastic calculus and to explore its connection with other areas such as partial differential equations (PDEs).

• Crash review of probability spaces and measures, theory of integrals, convergence Theo-rems; convergences of sequences of measurable functions;
• Radon-Nikodym Theorem; Conditional expectations;
• Martingales, submartingales and filtrations, submartingale inequality, upcrossings and down-crossings inequalities, submartingale convergence theorem, stopping and optional times and the Optional Sampling Theorem, Doob-Meyer Decompositions and quadratic variation pro-cesses;
• Constructions of Brownian motions, finite dimensional distributions, Kolmogorovs Con-sistency Theorem, Kolmogorovs Continuity Theorem, weak convergence, tightness and the Wiener measure, Levys modulus of continuity of Brownian motions;
• Constructions of stochastic integrals, local martingales and localizations, Ito's formula, Girsanov Theorem, Burkholder-Davis-Gundy inequality;
• Stochastic differential equations, existence and uniqueness of strong solutions, Markov property, strong Markov property, Markovian semigroups and infinitesimal generators, Feynman-Kac formula, Fokker-Planck operators and forwards and backwards Kolmogorov equations, existence and estimates of probability density;
• Malliavin calculus: an introduction.

Subject-specific Knowledge:

• By the end of the module students will:
• Be able to solve novel and/or complex problems in the field of Stochastic Analysis.
• Have an understanding of specialised and complex theoretical mathematics in the field of Stochastic Analysis.
• Have mastered a coherent body of knowledge of these subjects demonstrated in the following topic areas: measure and integrations; various convergence theorems of integrations and notions of variety of different convergence of sequence of measurable functions; conditional expectations and martingales; Brownian motions, stochastic integrals and stochastic differential equations; Markov property, Chapman-Kolmogorov equations and Fokker-Planck equations.
• Be able to reproduce theoretical mathematics related to this course at a level appropriate for Level 4, including key definitions and theorems.

Subject-specific Skills:

• Students will have developed advanced technical and scholastic skills in the area of Stochastic Analysis.

Key Skills:

• Students will have highly specialised skills in the following areas: problem solving, abstract reasoning.

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

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