Lecturer

Dr. Georgiy Shevchenko

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Time and Place

Lecture (March 12th to March 19th, 2014)

Tuesday, 18.03.2014, 14-16 in He220Wednesday, 19.03.2014, 14-16 in He220Friday, 21.03.2014, 14:30-16:30 in He220

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Prerequisites

Stochastics II

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Intended Audience

Master WiMa, Master of Finance, Master in Mathematics

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Fractional processes play an important role in modeling the long-range dependence property. In view of this, they find numerous applications, most notably in financial mathematics, geophysics, hydrology, telecommunication research etc. The most popular among fractional processes is the Gaussian one, viz. the fractional Brownian motion. However, it has several drawbacks. Firstly, its increments are stationary, which does not allow to model processes whose properties vary essentially as time flows. Secondly, it is self-similar, so has similar properties on different time scales, yet only few real-world processes have this property. Lastly, it has Gaussian distribution, so its tails are extremely light.In view of these drawbacks, various generalizations of fractional processes are studied in the literature. To address the first two drawbacks, it is useful to introduce some multifractional processes, by letting the memory parameter to vary with time. On this way, some multifractional counterparts of fractional Brownian motion were proposed. These processes are also Gaussian, so they do not solve the light tails problem. The third drawback can be removed by considering fractional and multifractional processes which have heavier tails than that of the Gaussian distribution. In this course two kinds of processes will be considered: stable processes (having heavy tails) and square Gaussian processes (having intermediate tails).

Contents

• Fractional Brownian motion: definition and basic properties.
• Multifractional Brownian motions(s).
• The localisability property.
• Double Wiener-Itô integrals.
• Non-central limit theorems and Rosenblatt process.
• Multifractional Rosenblatt process (mRp).
• Distributional properties of mRp.
• Pathwise properties of mRp: continuity, Hölder exponents, localisability.
• Square Gaussian random fields and their properties.
• Stable random variables and processes.
• LePage representation.
• Fractional and multifractional stable processes.
• Comparison to the Gaussian and the square Gaussian cases.
• Distributional and pathwise properties of multifractional stable processes.
• Fractional and multifractional stable fields.
• Local times and occupation densities of stochastic processes.
• Essence of the analytic method.
• General criteria for existence of local times.
• Conditions for the local time regularity in the Gaussian and stable cases.
• The local non-determinism property and its role in geometric properties of a random field.
• Local times of multifractional processes and their regularity.
• Open challenges.