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
| Date | Speaker | Content |
|---|---|---|
| 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
| Date | Speaker | Content |
|---|---|---|
| 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 |
| 16/03/2016 |
References:
[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.
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