Workshop Programme

Bar Ilan University:

• Arthur Yosef The Set-Indexed Brownian Motion
• Ely Merzbach (joint work with E. Herbin) The Set-Indexed Fractional Brownian Motion
Abstract
We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independence is relaxed. Relations with the Levy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motion. Behavior of the set-indexed fractional Brownian motion along increasing paths is analysed. Finally, regularity conditions are exhibited.  


• Elina Moldavskaya Regression models with long-range dependence and constraints
Abstract
We consider linear regression models with long-range dependence (LRD) in the noise and inequality-constraints on the parameters. We examine the solution of minimization problem of the least squares functional for such models. It is proved that this solution (least squares estimator) converges in distributions to the solution of the quadratic programming problem. The latter solution is non-Gaussian in typical cases in contrast to known results for models with LRD without constraints for which least squares estimator is asymptotically Caussian in many typical cases. Approximate representation for least squares estimator is given for particular cases. From this representation one can see the concrete structure of the estimators. 


• Yair Y. Shaki (joint work with Rami Atar and Adam Shwartz)  Fair Allocation between Large Server Pools
Abstract
I will discuss fairness towards servers in a parallel server model with many servers, in the heavy traffic diffusion regime of Halfin and Whitt. In practice one often uses the longestidle server first (LISF) routing policy, which attempts to equalize idleness among servers. This policy has a big advantage of working blindly, not requiring any information on model parameters. However, no analysis of this policy is available in the literature. The model we study has a fixed, arbitrary number of pools, and arbitrary (not necessarily equal) service rates within pools. Under mild assumptions we show that a LISF-like policy, where idleness is measured at the pool level, achieves fairness across server pools, in the limit.


• Boris Kriheli Wiener Integration with respect to Brownian Motion
Abstract
In this talk we give definitions and discuss properties of the following topics: a. The fractional integrals and derivatives both for finite and infinite intervals. Calculate the values of some important fractional derivatives. b. Fractional Brownian motion (fBm) and its spectral representation. The Mandelbrot - van Ness representation of fBm via the Wiener Process and some fractional kernels on real axes. c. Wiener integral with respect to fBm. d. Completeness of the Gaussian spaces generated by fBm in connection with their norms. e. Representation of fBm via the Wiener process on any finite interval and some representations for auxiliary processes. f. Moment estimates for Wiener integrals with respect to fBm.

Cardiff University:

• Stuart Petherick (joint work with N.Leonenko and A.Sikorskii) Fractal Activity Time Risky Asset Models
  
Abstract
The classical Geometric Brownian motion model for the price of a risky asset, from which the huge financial derivatives industry has developed, stipulates that the log returns are independently and identically distributed Gaussian. However, typical log returns data show a distribution with much higher peaks and heavier tails than the Gaussian, as well as evidence of long-range dependence. As a replacement, we consider a subordinator model based on fractal activity time. By the construction of this fractal activity time, we obtain separate models which lead to more flexible distributions for log returns such as Variance Gamma and Normal Inverse Gaussian. The construction is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Levy noise. The models we describe have desired features supported by real financial data. 

Kyiv University:

• Yuliya Mishura Diffusion Processes with Long Memory
Abstract
By an analogy to usual diffusion process, it is natural to define a diffusion process with long memory as a solution to a stochastic differential equation involving fBm. The mixed model involving both standard and fractional Brownian motion is even more flexible and is under consideration now inside the various problems. The definition, however, has several caveats, because, firstly, there are different versions of a stochastic integral with respect to fBm, secondly, when we also consider mixed processes, then the resulting solution must be integrable with respect to all the components. We discuss these and other problems related to diffusion processes with long memory.

• Yuriy Kozachenko Conditions and the Rate of Convergence of Expansions of Random Processes in Systems Generated by Wavelets
• Alexander Kukush Static Lower Bounds for Basket Payoffs and Comonotonic Distributions
Abstract
• Lyudmyla Sakhno On the Estimation of Higher-order Spectral Functionals
Abstract
The estimation of spectral functionals is relevant to many statistical problems in nonparametric and parametric statistics. These functionals can be used to represent some characteristics of stochastic processes and fields in nonparametric setting. On the other hand, these functionals appear in parametric estimation in spectral domain, e.g., when so-called minimum contrast (or quasi-likelihood) estimates are studied. We consider different estimators for the spectral functionals and their asymptotic behavior.

• Georgiy Shevchenko (joint work with M. Dozzi) Multifractional Harmonizable Stable Processes
Abstract
Whereas in the Gaussian case the different representations of fractional Brownian motion (fbm) are equivalent, the fractional stable processes which can be obtained from these representations are different. Also, when replacing the Hurst parameter in the representations of fbm by a Hurst function, we get different multifractional Gaussian processes. Therefore the class of multifractional stable processes (msp) is expected to be fairly rich, but caracterisations of this class of processes are not yet known. We give some results on the local time of the harmonisable msp, especially the regularity properties of its trajectories and limit theorems for the occupation integrals.

• Victoria KnopovaTransition Probability Density Estimates for Lévy and Lévy-type Processes
Abstract
The talk is devoted to the transition density estimates of a Lévy process with analytic characteristic function. Using the complex analysis technique (in particular, the steepest descent method), we construct the off-diagonal estimates for the transition probability density in terms of the carré du champ operator related to the process.

• Kostiantyn Ralchenko (joint work with G. Shevchenko) Absolute Continuous Approximations for Solutions of SDEs with fBm
Abstract
As it was mentioned in the papers (Androshchuk 2005, Androshchuk, Mishura 2006), the stochastic integral with respect to fBm can be approximated with Lebesgue-Stieltjes integral and the rate of convergence was established. We develop this approach and consider the approximation of solutions to path-wise stochastic differential equations with fBm by solutions of random ordinary differential equations.

Nancy University:

• Antoine Lejay Short Course on Rough Paths
Abstract
The theory of rough path has emerged recently as a tool to study generalization of stochastic differential equations (SDE) driven by processes other than Brownian motion and semi-martingales, as well as a tool to get a deeper understanding of some of the properties of SDEs. The goal of the short lecture is to give an insight of the theory as well as the core elements and the main properties of the rough paths and rough path integration.