IS15: Extremes of Gaussian and Related Random Fields

Organizer: Enkelejd Hashorva (University of Lausanne)

Asymptotic Behavior of Path Functionals for Vector-Valued Gaussian Processes at High Levels

Pavel Ievlev (University of Lausanne)

Understanding how long a stochastic system stays in a “safe” region is a core question in risk management, queueing and reliability. In this project we study high exceedence probabilities of the form \[\mathbb{P} \{ \Gamma_{[0,T]} ( \hat{\boldsymbol{u}} ( \boldsymbol{X} - u \boldsymbol{b} ) ) > L_u \},\] as \(u \to \infty\), where \(\Gamma_{[0,T]}\) is a functional of a continuous \(d\)-dimensional Gaussian process \(\mathbf X(t)\) on \([0,T]\), and \(L_u\) is some sequence of thresholds, chosen appropriately for each \(\Gamma\). The class of functionals we treat is quite broad, including functionals of the form \[\Gamma_E ( \boldsymbol{f} ) = \int_{E} G ( \boldsymbol{f} ( t ) ) \, d t \quad \text{and} \quad \Gamma_{E \times F} ( \boldsymbol{f} ) = \sup_{t \in E} \inf_{s \in F} \min_{i = 1, \dots, d} f_i ( t, s ),\] where \(G\) is some function satisfying additional assumptions. In particular, this class includes the classical sojourn time, Parisian (moving-window infimum) functional, area under the curve, as well as compositions of those with continuous but not necessarily linear operators. Regarding the class of the Gaussian processes, we study both stationary and non-stationary cases under the assumptions similar to those of Dȩbicki-Hashorva-Wang (2019).

Key technical contributions include the extension of Pickands-type arguments to these vector-valued settings and general functionals, supported by lemmas detailing conditional process behavior, uniform convergence, and properties of the functionals themselves. The presentation will outline the main theorems, discuss the crucial assumptions, and illustrate the framework’s applicability with examples. This work provides a unified approach to understanding extreme sojourns for a broad class of Gaussian models.

Hitting probabilities for multivariate Brownian motion: exact asymptotics

Svyatoslav Novikov (University of Lausanne)

Consider the multivariate Brownian risk model driven by the \(d\)-dimensional Brownian motion \(B(t), t\geq 0\). Classical results in the literature are concerned with the approximation of the simultaneous ruin probability, which is related to the approximation of the supremum of the process falling on certain regular sets \(E_u=(u,\infty)^d\). In this talk bounds and exact asymptotics of the probability \(\mathbb{P}\{\exists_{t \in [0,T]}: GB(t)-ct \in E_u\}\) for some set \(E_u\) are discussed. While \(G\) is a fixed non-singular matrix, the set \(E_u\) of \(\mathbb{R}^d\) can be quite general and varies with the threshold \(u\). Our examples include balls and parabolic shapes.

A new technical aspect of this talk is the connection with the results on Wiener sausage.

Jointly with Krzysztof Debicki and Nikolai Kriukov