# Fractional Calculus and Fractional Stochastic Calculus, 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.

### Yulia Mishura,

Invited Lecturer

Fractional Calculus and Fractional Stochastic Calculus with Applications

examples.

Session 1 solutions to exercises

### Elena Boguslavskaya

On some connections between martingales and fractional derivatives

### Vassili Kolokoltsov

Probabilistic point of view on generalized fractional calculus and generalized operator-valued Mittag-Leffler functions.

### Nikolai Leonenko

Most likely location of paths for a class of Feller processes

### Hao Ni

Modelling the effects of data streams using the Logsig-RNN model

### Enrico Scalas

## Invited Lecturer

### Yulia Mishura

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.

## Supporting Lecturers

### Elena Boguslavskaya

Brunel University London

### Vassili Kolokoltsov

University of Warwick

### Nikolai Leonenko

Cardiff University

### Jozsef Lorinczi

Loughborough University

### Hao Ni

UCL

### Enrico Scalas

University of Sussex

## Supported by London Mathematical Society

This event is part of the LMS Invited Lectures Series

## Get in Touch

Kingston Ln, London, Uxbridge UB8 3PH, UK