High Excursions of Gaussian Nonstationary Processes in Discrete Time (original) (raw)
On the ruin probability for physical fractional Brownian motion
Stochastic Processes and their Applications, 2004
We derive the exact asymptotic behavior of the ruin probability P{X (t) ¿ x for some t ¿ 0} for the process X (t) = t 0 (s) ds − ct, with respect to level x which tends to inÿnity. We assume that the underlying process (t) is a.s. continuous stationary Gaussian with mean zero and correlation function regularly varying at inÿnity with index −a ∈ (−1; 0).
Methodology and Computing in Applied Probability, 2013
We state large deviations for small time of a pinned n-conditional Gaussian process, i.e. the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants, by letting all the past monitoring instants to depend on the small parameter going to 0. Differently from what already developed in Caramellino and Pacchiarotti [9], this procedure is able to catch the dependence on the past observations. We apply the results to numerical experiments that involve the fractional Brownian motion, for the computation of the hitting probability through Monte Carlo methods. Keywords. Conditioned Gaussian processes; reproducing kernel Hilbert spaces; large deviations; exit time probabilities; Monte Carlo methods.
Logarithmic L2-Small Ball Asymptotics for some Fractional Gaussian Processes
Theory of Probability & Its Applications, 2005
We find the logarithmic L 2 -small ball asymptotics of some Gaussian processes related to the fractional Brownian motion, fractional Ornstein -Uhlenbeck process (fOU) and their integrated analogues. To that end we use general theorems on spectral asymptotics of integral operators obtained by combining them with the classical theorem of Weyl. In the simplest case of fractional Brownian motion we generalize the result of . We consider also the fractional Lévy's Brownian motion as well as the multiparameter fOU process on the bounded domain.
arXiv (Cornell University), 2023
Let B H be a d-dimensional fractional Brownian motion with Hurst index H ∈ (0, 1), f : [0, 1] −→ R d a Borel function, and E ⊂ [0, 1], F ⊂ R d are given Borel sets. The focus of this paper is on hitting probabilities of the non-centered Gaussian process B H + f. It aims to highlight how each component f , E and F is involved in determining the upper and lower bounds of P{(B H + f)(E) ∩ F = ∅}. When F is a singleton and f is a general measurable drift, some new estimates are obtained for the last probability by means of suitables Hausdorff measure and capacity of the graph Gr E (f). As application we deal with the issue of polarity of points for (B H + f)| E (the restriction of B H + f to the subset E ⊂ (0, ∞)).
Large deviations of infinite intersections of events in Gaussian processes
Stochastic Processes and their Applications, 2006
Consider events of the form {Z s ≥ ζ (s), s ∈ S}, where Z is a continuous Gaussian process with stationary increments, ζ is a function that belongs to the reproducing kernel Hilbert space R of process Z , and S ⊂ R is compact. The main problem considered in this paper is identifying the function β * ∈ R satisfying β * (s) ≥ ζ (s) on S and having minimal R-norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ (s) = s for s ∈ [0, 1] and Z is either a fractional Brownian motion or an integrated Ornstein-Uhlenbeck process.
Hitting times for Gaussian processes
The Annals of Probability, 2008
We establish a general formula for the Laplace transform of the hitting times of a Gaussian process. Some consequences are derived, and particular cases like the fractional Brownian motion are discussed.
Large deviations for excursions of non-homogeneous Markov processes
Electronic Communications in Probability, 2014
In this paper, the large deviations at the trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the twodimensional lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (downward jumps) depend on the position of the process-the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (upward jumps) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes converge to 0. The main result is that the probabilities of excursions far from 0 tend to 0 exponentially fast with an exponent proportional to the square of the scaling parameter. The proportionality coefficient is an integral of a linear combination of path components. A rate function of the large deviation principle is calculated for continuous functions only.