Fractional Calculus, Fractional Stochastic Calculus, including Rough-Paths, with Applications
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.
Lecture Series Program
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.
Some connections between martingales and fractional derivatives
Probabilistic methods in the analysis of
Multifractal stochastic processes
Properties of solutions of non-local equations
Modelling the effects of data streams using rough paths theory
Notes on a fractional non-homogeneous Poisson process
University of Kyiv
Yuliya 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.
Brunel University London
University of Warwick
University of Sussex
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