Conference Agenda

We consider spherical fractional Brownian motion $(S_H(\eta))_{\eta\in\mathbb{S}_{d-1}}$, which is obtained by taking fractional Brownian motion indexed by the (multi-dimensional) sphere $\mathbb{S}_{d-1}$, and calculate its persistence exponent. Persistence in this context is the study of the decay of the probability $$ \mathbb{P}\left( \sup_{\eta \in \mathbb{S}_{d-1}} S_H(\eta) \leq \varepsilon \right) $$ when the barrier $\varepsilon \searrow 0$ becomes more and more restrictive. Our main result shows that the persistence probability of spherical fractional Brownian motion has the same order of polynomial decay as its Euclidean counterpart.

We re-consider a multivariate fractional Brownian motion with component-wise different Hurst exponents. We investigate how strongly processes with different autocorrelation patterns over time can be correlated. We highlight that the multivariate model facilitates efficiency gains in forecasting compared to using univariate fractional Brownian motions only. This is demonstrated to be practically relevant for multivariate time series of realized volatilities. Therefore, we can improve the prediction of rough volatility. We advance the statistical theory for the model to provide parameter estimates, asymptotic confidence intervals and hypothesis tests, which are of interest for the application. We show that correlations do not only help to reduce prediction uncertainty, but moreover can be exploited to minimize statistical risk.

In this presentation, a method to exactly sample the trajectories of inverse subordinators, jointly with the undershooting and overshooting processes, will be provided. The method is applicable to general subordinators. To deal with such non-Markovian processes, we use the theory of semi-Markov processes, and recent developments for exact simulation of first passage events (i.e. passage time, undershooting, and overshooting) at a single time, developed by Cázares, Lin, and Mijatović (2023). To the best of our knowledge, the presented method is the first one that exactly simulates the whole trajectory of a general inverse subordinator.

Additionally, the Monte Carlo approximation of a functional of subdiffusive processes (in the form of time-changed Feller processes) will be considered, where a central limit theorem and the Berry-Esseen bounds will be presented. The approximation of time-changed Itô diffusions is also studied where the strong error is explicitly evaluated as a function of the time step, demonstrating the strong convergence.

The standard telegraph process $X_t$ describes the random motion of a particle on the real line with velocity changing alternately between a positive and a negative value ($\pm c$, $c>0$). Classically, the sequence of changing epochs is governed by a homogeneous Poisson process $N_t$ with rate $\lambda>0$. This means that the random times between consecutive reversals of direction of the motion are independent, identically distributed and exponentially distributed. Due to the important role of the telegraph process in various applied contexts, several generalizations of such a process have been studied. Since the memoryless property of the times between consecutive velocity changes represents a rare case in real phenomena, in some papers the sequence of changing epochs is supposed as governed by a non-homogeneous Poisson process with rate function $\lambda(t)>0$. In the latter case, the pde satisfied by the probability density function of the process is given by \begin{equation} \frac{\partial^2 p}{\partial t^2}+2 \lambda(t)\frac{\partial p}{\partial t}=c^2\frac{\partial^2 p}{\partial x^2},\quad x\in {\mathbb R},\quad t>0. \qquad (1) \end{equation} When $\lambda(t)=t^{-1}$, Eq. (1) identifies with the well-known one-dimensional Euler-Poisson-Darboux (EPD) equation, which, under suitable initial conditions, admits an explicit solution (see also [3]).

We study a modification of the Euler-Poisson-Darboux equation. Specifically, we consider a time-decreasing intensity function that tends to $0$ more slowly with respect to $\lambda(t)=\frac{\alpha}{t}$, in order to describe a random motion characterized by a long tail behaviour, typical of several physical, biological, economic and financial behaviour (see, for example [1]). Therefore, we assume that $N_t$ is a non-homogeneous Poisson process with rate function $\lambda(t)=\frac{\alpha}{\sqrt{t}}$, $\alpha>0$, $t>0$. Following the lines of [2], we consider the Fourier transform of the related pde (1) and solve it by means of a suitable transformation which leads to a one-dimensional Schr$\ddot{o}$dinger-type ordinary differential equation. The solution of such equation, which, to the best of our knowledge, is unknown in the literature, is obtained by means of the Frobenius method and provides a closed form expression of the characteristic function of $X_t$. Differently from the EPD equation, in such modified Euler-Poisson-Darboux equation, we have that $\int_{0}^{t} \lambda(s) {\rm d}s<+\infty$, so that the probability law admits also a discrete component. Starting from the characteristic function, we obtain the expression of the $n$-th moment of $X_t$, $n\in {\mathbb N}$, which is expressed as a finite sum of modified spherical Bessel functions. We determine Kac's-type scaling conditions under which the characteristic function of the non-homogeneous telegraph process $X_t$ tends to the one of the fractional Brownian motion with Hurst index $H=3/4$. Moreover, we study the probability law of $X_t$. This is composed by an absolutely continuous component on $(-c t, c t)$ and a discrete one concentrated on $c t$. The study of the behavior of the density at the end points of the interval $(-c t, c t)$ is finally provided.

[1] F. den Hollander, Long Time Tails in Physics and Mathematics. In: Grimmett, G. (eds) Probability and Phase Transition. NATO ASI Series, vol 420. Springer, Dordrecht, (1994).

[2] S.K. Foong, U. van Kolck, Poisson Random Walk for Solving Wave Equations, Prog. Theor. Phys. 87 (2) (1992) 285--292.

[3] R. Garra, E. Orsingher, Random flights related to the Euler-Poisson- Darboux equation, Markov Process. Relat. Fields. 22 (1) (2016) 87--110.

[4] B. Martinucci, S. Spina, On a finite-velocity random motion governed by a modified Euler-Poisson-Darboux equation, submitted.