Page 562 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 562
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)
Levy moving average models and present techniques from Malliavin calculus on Poisson spaces
to get quantitative versions associated with normal limits.
Prabhakar fractional operators and some related stochastic processes
Federico Polito, federico.polito@unito.it
University of Torino, Italy
The aim of this talk is to present properties and relevant definitions of some stochastic pro-
cesses related to generalized Mittag-Leffler functions, also known as Prabhakar functions. We
first describe a generalization of classical fractional calculus in which integral and differential
operators involve generalized Mittag-Leffler functions as kernels, highlighting especially how
differential operators should be regularized in order to turn them into Caputo-like operators.
Then, we analyze two renewal processes both based on generalized Mittag-Leffler functions in
two different manners. Lastly, we end the talk by characterizing time-changed Lévy processes
whose governing equations present generalized Mittag-Leffler fractional operators.
References
[1] Renewal processes based on generalized Mittag-Leffler waiting times (with DO Cahoy).
Communications in Nonlinear Science and Numerical Simulation, Vol. 18 (3), 639-650,
2013.
[2] Hilfer-Prabhakar Derivatives and Some Applications (with R Garra, R Gorenflo and Z
Tomovski). Applied Mathematics and Computation, Vol. 242, 576-589, 2014.
[3] Fractional Diffusion-Telegraph Equations and their Associated Stochastic Solutions (with
M D’Ovidio). Theory of Probability and its Applications, Vol. 62 (4), 692-718, 2017 (En-
glish version: Vol. 62 (4), 552-574, 2018).
[4] A practical guide to Prabhakar fractional calculus (with A Giusti, I Colombaro, R Garra,
R Garrappa, M Popolizio and F Mainardi). To appear on Fractional Calculus and Applied
Analysis, 2020.
On simulation of rough Volterra stochastic volatility models
Jan Pospíšil, honik@kma.zcu.cz
University of West Bohemia, Czech Republic
Coauthor: Jan Matas
Rough Volterra volatility models are a progressive and promising field of research in derivative
pricing. Although rough fractional stochastic volatility models already proved to be superior
in real market data fitting, techniques used in simulation of these models are still inefficient in
terms of speed and accuracy. This talk aims to present the accurate tools and techniques that
could be used also in nowadays largely emerging pricing methods based on machine learning.
In particular we compare three widely used methods: the Cholesky method, Hybrid scheme
and the rDonsker scheme for simulation of the rBergomi model and for a more general αRFSV
model. We also comment on implemention of variance reduction techniques, especially we
560
PROCESSES (MS-18)
Levy moving average models and present techniques from Malliavin calculus on Poisson spaces
to get quantitative versions associated with normal limits.
Prabhakar fractional operators and some related stochastic processes
Federico Polito, federico.polito@unito.it
University of Torino, Italy
The aim of this talk is to present properties and relevant definitions of some stochastic pro-
cesses related to generalized Mittag-Leffler functions, also known as Prabhakar functions. We
first describe a generalization of classical fractional calculus in which integral and differential
operators involve generalized Mittag-Leffler functions as kernels, highlighting especially how
differential operators should be regularized in order to turn them into Caputo-like operators.
Then, we analyze two renewal processes both based on generalized Mittag-Leffler functions in
two different manners. Lastly, we end the talk by characterizing time-changed Lévy processes
whose governing equations present generalized Mittag-Leffler fractional operators.
References
[1] Renewal processes based on generalized Mittag-Leffler waiting times (with DO Cahoy).
Communications in Nonlinear Science and Numerical Simulation, Vol. 18 (3), 639-650,
2013.
[2] Hilfer-Prabhakar Derivatives and Some Applications (with R Garra, R Gorenflo and Z
Tomovski). Applied Mathematics and Computation, Vol. 242, 576-589, 2014.
[3] Fractional Diffusion-Telegraph Equations and their Associated Stochastic Solutions (with
M D’Ovidio). Theory of Probability and its Applications, Vol. 62 (4), 692-718, 2017 (En-
glish version: Vol. 62 (4), 552-574, 2018).
[4] A practical guide to Prabhakar fractional calculus (with A Giusti, I Colombaro, R Garra,
R Garrappa, M Popolizio and F Mainardi). To appear on Fractional Calculus and Applied
Analysis, 2020.
On simulation of rough Volterra stochastic volatility models
Jan Pospíšil, honik@kma.zcu.cz
University of West Bohemia, Czech Republic
Coauthor: Jan Matas
Rough Volterra volatility models are a progressive and promising field of research in derivative
pricing. Although rough fractional stochastic volatility models already proved to be superior
in real market data fitting, techniques used in simulation of these models are still inefficient in
terms of speed and accuracy. This talk aims to present the accurate tools and techniques that
could be used also in nowadays largely emerging pricing methods based on machine learning.
In particular we compare three widely used methods: the Cholesky method, Hybrid scheme
and the rDonsker scheme for simulation of the rBergomi model and for a more general αRFSV
model. We also comment on implemention of variance reduction techniques, especially we
560
