Topic: Fractional Brownian motion and related topics

Location: Room 315, Sherfield Building (see details below)
Date & Time: Wednesdays, 1pm - 2.30pm
Initiators & Supervisors: Dr. Antoine Jacquier and Dr. Mikko Pakkanen 
Organiser: Maxime Morariu-Patrichi
Syllabus: download here 

For any questions, please feel free to contact the organiser.

Room 315 is located on the 3rd floor in the Sherfield building. The room is in the Centre for Academic English area. Follow the arrows that lead to Room 319 of the Centre for Academic English.

Autumn Term

28/10/2015   Planning the reading group for the autumn term
4/11/2015 Andrea Granelli Introducing fractional Brownian Motion
Definition of fractional Brownian motion (fBM), Gaussian process viewpoint, self-similarity property, long range dependence of increments, non-Markov property, non-semimartingale property
11/11/2015 Martin Weidner Integral representations and reproducing kernels
Origins of the name 'fractional Brownian motion', definition of fractional integrals and derivatives, Mandelbrot Van Ness integral representation of fBM,  reproducing kernel Hilbert space, Cameron-Martin theorem, Golosov integral representation, spectral integral representation for the case of an infinite time horizon
18/11/2015 Sergey Badikov Further properties and characterisations
Lévy characterisation of fBM, maximal inequality theorem for deterministic times, maximal inequality theorem for stopping times
25/11/2015 Douglas Machado Vieira Local times
Local times for Brownian motion, review of tempered distributions, Donsker delta distribution, local times for fBM, weighted local times, pricing call options under geometric fBM
Download the slides here
2/12/2015 Michael Chau Pathwise integration
Hölder continuity, p-variation norm, Young's integral and pathwise integration, Fubini's thoerem
9/12/2015 Maxime Morariu-Patrichi Itô calculus and Girsanov theorem
Itô formula for functions of linear combination of fBMs with H>=1/2, Ito formula with weaker conditions than the twice continuous differentiability, Molchan martingale, changes of measure and Girsanov's theorem


Spring Term

27/01/2016   Planning the reading group for the spring term
3/02/2016 Claudio Heinrich SDEs driven by fBM
Reminder for the case H=1/2, fractional Besov type spaces, existence and uniqueness for H>1/2, existence of moments, Euler approximation scheme and convergence
Download the slides here
10/02/2016 Henry Stone Simulation methods for fMB
Hosking method, Cholesky method, Davis and Harte method, integral representation methods (Mandelbrot Van Ness, hypergeometric function), spectral method, simulation with micropulses
17/02/2016 Nengli Lim  A Stratonovich-Skorohod integral formula for Gaussian rough paths
Download the slides here 
24/02/2016 Arman Khaledian The arbitrage issue
9/03/2016 Maxime Morariu-Patrichi  Volatility is rough


[1] Y. Mishura. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, Springer, 2010.

[2] S. Rostek. Option pricing in Fractional Brownian markets. Lecture Notes in Economics and Mathematical Systems, 2009.

[3] F. Biagini, Y. Hu, B. Oksendal and T. Zhang. Stochastic Calculus for Fractional Brownian Motion and Applications. Probability and its Applications, 2008.

[4] L. Decreusefond and A.S. Ustunel. Stochastic analysis of the fractional Brownian motion. Potential Analysis, 10: 177-214, 1999.

[5] C. Jost. Integral Transformations of Volterra Gaussian Processes. PhD Thesis (introduction), University of Helsinki, 2007.

[6] G. Samorodnitsky and M. Taqqu. Stable Non-Gaussian Random Processes. New York, Chapman and Hall, 1994.