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Fractional Calculus and Fractional Stochastic Calculus, including Rough-Paths, with Applications with Professor Yulia Mishura

Yuliya Mishura

Supported by London Mathematical Society

This event is part of the LMS Invited Lectures Series

 

There are two distinct prominent topics in the theory of stochastic processes: the theory of fractional processes (including stochastic differential equations and stochastic partial differential equations with respect to fractional processes), and the theory of stochastic differential equations and stochastic partial differential equations with fractional operators. These topics are connected via fractional calculus. Fractional calculus is deterministic (not stochastic), and is simply a specific set of tools and methods. However, mastering these methods allows one not only to consider the analytical properties of the solutions of the above equations, but also to apply them to problems in physics, mechanics, functioning of communication systems, finance, economics, statistics, and many other applications.


The proposed course of lectures is precisely aimed at explaining the mathematical arguments of fractional calculus and fractional stochastic calculus, including rough-path theory, with subsequent application to stochastic differential equations (SDEs), stochastic partial differential equations (SPDEs), and finance. It will consider fractional processes as well as the theory of stochastic differential equations and stochastic partial differential equations with fractional operators, and study the connection between the two, with application to finance.
This course is aimed at a broad mathematical audience interested in working with long- and short-memory processes. In particular it will be beneficial for PhD students. The course will be accompanied by exercise sessions.

 Schedule TBC

 

 Maths seminar

 

Invited Lecturer

 

Professor Yulia Mishura - University of Kyiv

Professor Mishura has been working in Taras Shevchenko National University of Kyiv, Ukraine since 1976, becoming a professor in 1991. She has been the head of the Department of Probability, Statistics and Actuarial Mathematics since 2003. She is also the head of the research group “Exact formulas, estimates, asymptotic properties, and statistical analysis of complex evolutionary systems with many degrees of freedom."


At present the research interests of Y. Mishura lay in the area of the interplay between fractional stochastic processes and differential equations with respect to fractional operators, where random processes arise. However, the random processes which arise from solving fractional differential equations are very different from fractional stochastic processes. Luckily the fractional calculus happened to be the connecting bridge between them. The purpose of the proposed course is to present both an exposition of the basic ideas in the theory of fractional stochastic processes and the main ideas from the theory of differential equations with fractional operators. Both these areas at the moment are very popular and are rapidly developing, speeded up by a large number of applications.

  

Fractional Calculus and Fractional Stochastic Calculus with Applications

• Elements of fractional calculus. Fractional differential and fractional partial differential operators. Stable processes. Stochastic differential equations with fractional differential and fractional partial differential operators. Introduction to the classes of solutions and the main properties of the solutions.

• Fractional Gaussian processes and their main properties. Elements of rough-path theory. Stochastic differential equations involving fractional processes. Existence-uniqueness results and the main properties of the solutions.

• Partial stochastic differential equations involving fractional processes. Description of the corresponding functional spaces and the properties of solutions. Parabolic equations involving the Wiener process and fractional Brownian motion as an example.

• Stochastic and partial stochastic differential equations with fractional operators: problems of stability and asymptotic behavior of solutions. Statistical parameter estimation in fractional stochastic differential equations. Estimation of the Hurst index, drift and diffusion parameters. Financial applications to the models with fractional stochastic volatility.

 

Supporting Lecturers 

 

Dr Elena Boguslavskaya - Brunel University London

Some connections between martingales and fractional derivatives

 

Dr Vassili Kolokoltsov - University of Warwick

Probabilistic methods in the analysis of fractional PDEs

 

Dr Nikolai Leonenko - Cardiff University

Multifractal stochastic processes

 

Dr Jozsef Lorinczi - Loughborough University

Properties of solutions of non-local equations

 

Dr Hao Ni - UCL

Modelling the effects of data streams using rough paths theory

 

Dr Enrico Scalas - University of Sussex

Notes on a fractional non-homogeneous Poisson process

 

London Mathematical Society Invited Lecture Series 

The annual Invited Lecturers Series aim to bring a distinguished overseas mathematician to the United Kingdom to present a small course of about ten lectures spread over a week. Each course of Invited Lectures is on a major field of current mathematical research, and is instructional in nature, being directed both at graduate students beginning research and at established mathematicians who wish to learn about a field outside their own research specialism.

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Registration

Register now through this form.

For further details about the LMS Invited Lecture Series please contact Dr Elena Boguslavskaya 

 

Travel Support

Funds available for partial support to attend the LMS Invited Lecture Series. Requests for support with an estimate of expenses should be addressed to the organiser:  Dr Elena Boguslavskaya