Fractional Brownian motion and related processes: stochastic calculus, statistical applications and modeling


Prof. Dr. Yuliya Mishura

Time and place


The course will be held from June 15 to June 25, 2015 on every

Monday 2-4 pm
Tuesday, 2-4 pm
Wednesday, 8-10 am

and takes place in Helmholtzstraße 22, Room E18.


Elementary probability course  ("Elementare Wahrscheinlichkeitsrechnung und Statistik") and Stochastics I.

Intended audience

Master students in "Mathematik", "Wirtschaftsmathematik" and students in "Master of Finance".


  • Introduction into the theory of the processes with long-range dependence. The elements of fractional calculus. Elements of the theory of Gaussian processes. Definition and the main properties of fractional Brownian motion. Economical and statistical background. Fractional Brownian motion as the process with long-range dependence. How to model its trajectories. Self-similar processes. Index of self-similarity.
  • Statistical problem of parameter estimation in the models with long-range dependence with discrete and continuous time. Estimators for Hurst index as the index of self-similarity.
  • Linear and general diffusion models with fractional Brownian motion. Estimators of the unknown parameters of drift and volatility by the unique trajectory.
  • Asymptotic properties of statistical estimators in the models with long-range dependence. Simulation. Some elements of high-frequency data.
  • Stochastic differential equations with fractional Brownian motion. How to approximate the solutions and model their trajectories.
  • Modeling and simulations for the functionals of the fractional Brownian motion and their statistical applications. The notion of multifractionality and how to deal with multifractional processes.


Either a written or an oral examination, dependent on the number of participants.

Credit points: 2






Prof. Yuliya Mishura
Office hours: on appointment


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