arXiv subjects
Publications and source records attributed to Vladimir I. Piterbarg.
Asymptotic behavior of the point process of high and medium values of a Gaussian stationary process with discrete time is considered. An approximation by a Poisson cluster point process is given for the point process.
Gaussian random fields on finite dimensional smooth manifolds whose variances reach their maximum value at smooth submanifolds are considered. Exact asymptotic behaviors of large excursion probabilities have been evaluated. Vector Gaussian processes, chi-square processes, Bessel process, fractional Bessel process, Bessel bridge are examples of application of this result.
Gaussian random fields on Euclidean spaces whose variances reach their maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximum of theirs trajectories have been evaluated using Double Sum Method under the widest possible conditions.
We consider the fractional Cox-Ingersoll-Ross process satisfying the stochastic differential equation (SDE) $dX_t = aX_t\,dt + σ\sqrt{X_t}\,dB^H_t$ driven by a fractional Brownian motion (fBm) with Hurst parameter exceeding $\frac{2}{3}$. The integral $\int_0^t\sqrt{X_s}dB^H_s$ is considered as a pathwise integral and is equal to the limit of Riemann-Stieltjes integral sums. It is shown that the fractional Cox-Ingersoll-Ross process is a square of the fractional Ornstein-Uhlenbeck process until the first zero hitting. Based on that, we consider the square of the fractional Ornstein-Uhlenbeck process with an arbitrary Hurst index and prove that until its first zero hitting it satisfies the specified SDE if the integral $\int_0^t\sqrt{X_s}\,dB^H_s$ is defined as a pathwise Stratonovich integral. Therefore, the question about the first zero hitting time of the Cox-Ingersoll-Ross process, which matches the first zero hitting moment of the fractional Ornstein-Uhlenbeck process, is natural. Since the latter is a Gaussian process, it is proved by the estimates for distributions of Gaussian processes that for $a<0$ the probability of hitting zero in finite time is equal to 1, and in case of $a>0$ it is positive but less than 1. The upper bound for this probability is given.
For a centered $d$-dimensional Gaussian random vector $ξ=(ξ_1,\ldots,ξ_d)$ and a homogeneous function $h:R^d\to R$ we derive asymptotic expansions for the tail of the Gaussian chaos $h(ξ)$ given the function $h$ is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of $h(ξ)$ and its density at infinity and then discuss possible extensions for some general $ξ$ with polar representation.
Let $X_H(t), t\ge 0$ be a fractional Brownian motion with Hurst index $H\in(0,1}$ and define a gamma-reflected process $W_\Ga(t)=X_H(t)-ct-\gammainf_{s\in[0,t]}\left(X_H(s)-cs \right)$, $t\ge0$ with $c>0,γ\in [0,1]$ two given constants. In this paper we establish the exact tail asymptotic behaviour of $\sup_{t\in [0,T]} W_γ(t)$ for any $T\in (0,\IF]$. Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.