MSC 2010: 26A33, 05C72, 33E12, 34A08, 34K37, 35R11, 60G22The fractional calculus (FC) is an area of intensive research and development. In a previous paper and poster we tried to exhibit its recent state, surveying the period of 1966-2010. The poster accompanying the present note illustrates the major contributions during the period 1695-1970, the "old history" of FC.

MSC 2010: 26A33, 05C72, 33E12, 34A08, 34K37, 35R11, 60G22In the last decades fractional calculus became an area of intense re-search and development. The accompanying poster illustrates the major contributions during the period 1966-2010.

Given a two-dimensional fractional multiplicative process ${\left(F_t\right)}_{t\in [0,1]}$ determined by two Hurst exponents $H_1$ and $H_2$, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0,1]$ by $F$ if and only if $H_1=H_2$.

We study actuarial methods of option pricing in a fractional Black-Scholes model with time-dependent volatility. We interpret the option as a potential loss and we show that the fair premium needed to insure this loss coincides with the expectation of the discounted claim payoff under the average risk neutral measure.

We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin...

A stochastic “Fubini” lemma and an approximation theorem for integrals on the plane are used to produce a simulation algorithm for an anisotropic fractional Brownian sheet. The convergence rate is given. These results are valuable for any value of the Hurst parameters $({\alpha }_1,{\alpha }_2)\in {]0,1[}^2,{\alpha }_i\ne \frac{1}{2}.$ Finally, the approximation process is iterative on the quarter plane ${ℝ}_{+}^2.$ A sample of such simulations can be used to test estimators of the parameters αi,i = 1,2.

We prove, by means of Malliavin calculus, the convergence in $L^2$ of some properly renormalized weighted quadratic variations of bi-fractional Brownian motion (biFBM) with parameters $H$ and $K$, when $H\<1/4$ and $K\in (0,1]$.

In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW model is useful in physics, to model diffusing particles. Its scaling limit is a time-changed process, whose densities solve an anomalous diffusion equation. This paper develops limit theory and governing equations for cluster CTRW, in which a random number of jumps cluster together into a single jump. The clustering introduces a dependence between the waiting times and jumps that significantly affects...

In this paper, using direct and inverse images for fractional stochastic tangent sets, we establish the deterministic necessary and sufficient conditions which control that the solution of a given stochastic differential equation driven by the fractional Brownian motion evolves in some particular sets K. As a consequence, a comparison theorem is obtained.

A linear-quadratic control problem with an infinite time horizon for some infinite dimensional controlled stochastic differential equations driven by a fractional Brownian motion is formulated and solved. The feedback form of the optimal control and the optimal cost are given explicitly. The optimal control is the sum of the well known linear feedback control for the associated infinite dimensional deterministic linear-quadratic control problem and a suitable prediction of the adjoint optimal system...

The naïve and the least-squares extrapolation are investigated in the fractional autoregressive models of the first order. Some explicit formulas are derived for the one and two steps ahead extrapolation.

Motivated by applications in queueing fluid models and ruin theory, we analyze the asymptotics of $\mathbb{P}(su{p}_{t\in [0,T]}({\sum }_{i=1}^n{\lambda }_i{B}_{H_i}\left(t\right)-ct)>u)$, where $B_{H_i}\left(t\right):t\ge 0$, i = 1,...,n, are independent fractional Brownian motions with Hurst parameters $H_i\in (0,1]$ and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of $mi{n}_{i=1,...,n}{H}_i$.

Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. In the case of multiparameter Gaussian random fields, Adler proved that these two concepts are connected under the assumption of increment stationarity property. The aim of this paper is to consider the case of Gaussian fields without any stationarity condition. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph...

Multistable processes, that is, processes which are, at each “time”, tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise Hölder exponent...