We consider an important subclass of self-similar, non-gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar...

We present three new identities in law for quadratic functionals of conditioned bivariate Gaussian processes. In particular, our results provide a two-parameter generalization of a celebrated identity in law, involving the path variance of a Brownian bridge, due to Watson (1961). The proof is based on ideas from a recent note by J.-R. Pycke (2005) and on the stochastic Fubini theorem for general Gaussian measures proved in Deheuvels et al. (2004).

Let $X=X\left(t\right),t\in {ℝ}^N$ be a Gaussian random field in ${ℝ}^d$ with stationary increments. For any Borel set $E\subset {ℝ}^N$, we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.

Let $\left\{W_t\right\}=\left\{{(X_{t^{\text{'}}}^{\text{'}},Y_t^{\text{'}})}^{\text{'}}\right\}$ be vector ARMA $(m,n)$ processes. Denote by ${\widehat{X}}_t\left(a\right)$ the predictor of $X_t$ based on $X_{t-a},X_{t-a-1},...$ and by ${\widehat{X}}_t(a,b)$ the predictor of $X_t$ based on $X_{t-a},X_{t-a-1},...,Y_{t-b},Y_{t-b-1},...$. The accuracy of the predictors is measured by ${\Delta }_X\left(a\right)=\text{E}[X_t-{\widehat{X}}_t\left(a\right)]{[X_t-{\widehat{X}}_t\left(a\right)]}^{\text{'}}$ and ${\Delta }_X(a,b)=\text{E}[X_t-{\widehat{X}}_t(a,b)]{[X_t-{\widehat{X}}_t(a,b)]}^{\text{'}}$. A general sufficient condition for the equality ${\Delta }_X\left(a\right)={\Delta }_X(a,a)]$ is given in the paper and it is shown that the equality ${\Delta }_X\left(1\right)={\Delta }_X(1,1)]$ implies ${\Delta }_X\left(a\right)={\Delta }_X(a,a)]$ for all natural numbers $a$.

The problem of completeness of the forward rate based bond market model driven by a Lévy process under the physical measure is examined. The incompleteness of market in the case when the Lévy measure has a density function is shown. The required elements of the theory of stochastic integration over the compensated jump measure under a martingale measure are presented and the corresponding integral representation of local martingales is proven.

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...

The indifference valuation problem in incomplete binomial models is analyzed. The model is more general than the ones studied so far, because the stochastic factor, which generates the market incompleteness, may affect the transition propabilities and/or the values of the traded asset as well as the claim’s payoff. Two pricing algorithms are constructed which use, respectively, the minimal martingale and the minimal entropy measures. We study in detail the interplay among the different kinds of...

A one-to-one correspondence between locally square integrable periodically correlated (PC) processes and a certain class of infinite-dimensional stationary processes is obtained. The correspondence complements and clarifies Gladyshev's known result [3] describing the correlation function of a continuous periodically correlated process. In contrast to Gladyshev's paper, the procedure for explicit reconstruction of one process from the other is provided. A representation of a PC process as a unitary...

La théorie des corps convexes a commencé à la fin du xixe siècle avec l’inégalité de Brunn, généralisée ensuite sous la forme de l’inégalité de Brunn-Minkowski-Lusternik, qui s’applique à des ensembles non convexes. Ce thème a depuis longtemps des contacts avec les problèmes isopérimétriques et avec des inégalités d’Analyse telle que les plongements de Sobolev. On développera quelques aspects plus récents des inégalités géométriques, dont certains sont liés à la technique du transport de mesure,...