arXiv subjects

Krzysztof Debicki

Publications and source records attributed to Krzysztof Debicki.

At least 19 recordsLinked to original sources

arXiv2024

Probability of entering an orthant by correlated fractional Brownian motion with drift: Exact asymptotics

For $\{B_H(t)= (B_{H,1}(t), \ldots, B_{H,d}(t))^\top,t\ge0\}$, where $\{B_{H,i}(t),t\ge 0\}, 1\le i\le d$ are mutually independent fractional Brownian motions, we obtain the exact asymptotics of $$ \mathbb P (\exists t\ge 0: A B_{H}(t) - μt >νu), \ \ \ \ u\to\infty, $$ where $A$ is a non-singular $d\times d$ matrix and $μ=(μ_1,\ldots, μ_d)^\top\in R^d$, $ν=(ν_1, \ldots, ν_d)^\top \in R^d$ are such that there exists some $1\le i\le d$ such that $μ_i>0, ν_i>0.$

math.PR↗

arXiv2021

Extremes of Gaussian random fields with non-additive dependence structure

We derive exact asymptotics of $$\mathbb{P}\left(\sup_{\mathbf{t}\in {\mathcal{A}}}X(\mathbf{t})>u\right),~ \text{as}~ u\to\infty,$$ for a centered Gaussian field $X(\mathbf{t}),~ \mathbf{t}\in \mathcal{A}\subset\mathbb{R}^n$, $n>1$ with continuous sample paths a.s., for which $\arg \max_{\mathbf{t}\in {\mathcal{A}}} Var(X(\mathbf{t}))$ is a Jordan set with finite and positive Lebesque measure of dimension $k\leq n$ and its dependence structure is not necessarily locally stationary. Our findings are applied to deriving the asymptotics of tail probabilities related to performance tables and chi processes where the covariance structure is not locally stationary.

math.PR↗

arXiv2021

Simultaneous ruin probability for multivariate gaussian risk model

Let $\textbf{Z}(t)=(Z_1(t) ,\ldots, Z_d(t))^\top , t \in \mathbb{R}$ where $Z_i(t), t\in \mathbb{R}$, $i=1,...,d$ are mutually independent centered Gaussian processes with continuous sample paths a.s. and stationary increments. For $\textbf{X}(t)= A \textbf{Z}(t),\ t\in\mathbb{R}$, where $A$ is a nonsingular $d\times d$ real-valued matrix, $\textbf{u}, \textbf{c}\in\mathbb{R}^d$ and $T>0$ we derive tight bounds for \[ \mathbb{P}\left\{\exists_{t\in [0,T]}: \cap_{i=1}^d \{ X_i(t)- c_i t > u_i\}\right\} \] and find exact asymptotics as $(u_1,...,u_d)^{\top}= (u a_1,..., ua_d)^\top$ and $u\to\infty$.

math.PR↗

arXiv2020

Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

We derive the exact asymptotics of \[ P\left( \sup_{t\ge 0} \Bigl( X_1(t) - μ_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - μ_2 s\Bigr)> u \right), \ \ u\to\infty, \] where $(X_1(t),X_2(s))_{t,s\ge0}$ is a correlated two-dimensional Brownian motion with correlation $ρ\in[-1,1]$ and $μ_1,μ_2>0$. It appears that the play between $ρ$ and $μ_1,μ_2$ leads to several types of asymptotics. Although the exponent in the asymptotics as a function of $ρ$ is continuous, one can observe different types of prefactor functions depending on the range of $ρ$, which constitute a phase-type transition phenomena.

math.PR↗

arXiv2019

Approximation of Supremum of Max-Stable Stationary Processes and Pickands Constants

Let $X(t),t\in \mathbb{R}$ be a stochastically continuous stationary max-stable process with Fréchet marginals $Φ_α, α>0$ and set $M_X(T)=\sup_{t \in [0,T]} X(t),T>0$. In the light of the seminal articles [1,2], it follows that $A_T=M_X(T)/T^{1/α}$ converges in distribution as $T\to \infty$ to $\mathcal{H}_Z^{1/α} X(1)$, where $\mathcal{H}_Z$ is the Pickands constant corresponding to the spectral process $Z$ of $X$. In this contribution we derive explicit formulas for $\mathcal{H}_Z$ in terms of $Z$ and show necessary and sufficient conditions for its positivity. From our analysis it follows that $A_T^β,T>0$ is uniformly integrable for any $β\in (0,α)$. Further, we discuss the dissipative Rosiński (or mixed moving maxima) representation of $X$. Additionally, for Brown-Resnick $X$ we show the validity of the celebrated Slepian inequality and obtain lower bounds on the growth of supremum of Gaussian processes with stationary increments by exploiting the link between Pickands constants and Wills functional. Moreover, we derive upper bounds for supremum of centered Gaussian processes given in terms of Wills functional, and discuss the relation between Pickands and Piterbarg constants.

math.PR↗

arXiv2019

Sojourn times of Gaussian processes with trend

We derive exact tail asymptotics of sojourn time above the level $u\geq 0$ $$ \mathbb{P}\left(v(u)\int_0^T \mathbb{I}(X(t)-ct>u)d t>x\right), \quad x\geq 0 $$ as $u\to\infty$, where $X$ is a Gaussian process with continuous sample paths, $c>0$, $v(u)$ is a positive function of $u$ and $T\in (0,\infty]$. Additionally, we analyze asymptotic distributional properties of $$τ_u(x):=\inf\left\{t\geq 0: v(u) \int_0^t \mathbb{I}(X(s)-cs>u)d s>x\right\}, $$ as $u\to\infty$, $x\geq 0$, where $\inf\emptyset=\infty$. The findings of this contribution are illustrated by a detailed analysis of a class of Gaussian processes with stationary increments and a family of self-similar processes.

math.PR↗

arXiv2019

Logarithmic asymptotics for probability of component-wise ruin in a two-dimensional Brownian model

We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function $P(u)$ for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where $u$ is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of $-\ln P(u)/u$ as $u$ tends to infinity, which depends essentially on the correlation $ρ$ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem.

math.PR↗

arXiv2018

Extremes of nonstationary Gaussian fluid queues

This contribution investigates asymptotic properties of transient queue length process $$ Q(t)=\max\left(x+X(t)-ct, \sup_{0\leq s\leq t}\left(X(t)-X(s)-c(t-s)\right)\right),\ \ \ t\geq 0 $$ in Gaussian fluid queueing model, where input process $X$ is modeled by a centered Gaussian process with stationary increments, $c>0$ is the output rate and $x=Q(0)\ge0$. More specifically, under some mild conditions on $X$, exact asymptotics of $$\mathbb{P}\left(Q(T_u)>u\right) $$ as $u\to\infty$, is derived. The play between $u$ and $T_u$ leads to two qualitatively different regimes: (A) short-time horizon when $T_u$ is relatively small with respect to $u$; (B) moderate- or long-time horizon when $T_u$ is asymptotically much larger than $u$. As a by-product, some implications for the speed of convergence to stationarity of the considered model are discussed.

math.PR↗

arXiv2018

Extremes of vector-valued Gaussian processes with Trend

Let $X(t)=(X_1(t), \dots, X_n(t)), t\in \mathcal{T}\subset \mathbb{R} $ be a centered vector-valued Gaussian process with independent components and continuous trajectories, and $h(t)=(h_1(t),\dots, h_n(t)), t\in \mathcal{T} $ be a vector-valued continuous function. We investigate the asymptotics of $$\mathbb{P}\left(\sup_{t\in \mathcal{T} } \min_{1\leq i\leq n}(X_i(t)+h_i(t))>u\right)$$ as $u\to\infty$. As an illustration to the derived results we analyze two important classes of $X(t)$: with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.

math.PR↗

arXiv2018

The time of ultimate recovery in Gaussian risk model

We analyze the distance $\mathcal{R}_T(u)$ between the first and the last passage time of $\{X(t)-ct:t\in [0,T]\}$ at level $u$ in time horizon $T\in(0,\infty]$, where $X$ is a centered Gaussian process with stationary increments and $c\in\mathbb{R}$, given that the first passage time occurred before $T$. Under some tractable assumptions on $X$, we find $Δ(u)$ and $G(x)$ such that $$\lim_{u\to\infty}\mathbb{P}\left(\mathcal{R}_T(u)>Δ(u)x\right)=G(x),$$ for $x\geq 0$. We distinguish two scenarios: $T<\infty$ and $T=\infty$, that lead to qualitatively different asymptotics. The obtained results provide exact asymptotics of the ultimate recovery time after the ruin in Gaussian risk model.

math.PR↗

arXiv2017

Extremes of $γ$-reflected Gaussian process with stationary increments

For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_γ(t)=X(t)-ct-γ\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $γ$-reflected process, where $γ\in (0,1)$. This process is introduced in the context of risk theory to model surplus process that include tax payments of loss-carry forward type.In this contribution we derive asymptotic approximations of both the ruin probability and the joint distribution of first and last passage times given that ruin occurs. We apply our findings to the cases with $X$ being the multiplex fractional Brownian motion and the integrated Gaussian processes. As a by-product we derive an extension of Piterbarg inequality \KD{for} threshold-dependent random fields.

math.PR↗

arXiv2015

On Parisian ruin over a finite-time horizon

For a risk process $R_u(t)=u+ct-X(t), t\ge 0$, where $u\ge 0$ is the initial capital, $c>0$ is the premium rate and $X(t),t\ge 0$ is an aggregate claim process, we investigate the probability of the Parisian ruin \[ \mathcal{P}_S(u,T_u)=\mathbb{P}\{\inf_{t\in[0,S]} \sup_{s\in[t,t+T_u]} R_u(s)<0\}, \] with a given positive constant $S$ and a positive measurable function $T_u$. We derive asymptotic expansion of $\mathcal{P}_S(u,T_u)$, as $u\to\infty$, for the aggregate claim process $X$ modeled by Gaussian processes. As a by-product, we derive the exact tail asymptotics of the infimum of a standard Brownian motion with drift over a finite-time interval.

math.PR↗

arXiv2014

Extremes of Order Statistics of Stationary Processes

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$ the $r$th largest order statistics of $X_1(t), \ldots , X_n(t), t\ge 0$. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions $C_{r,T,u}$ is not empty. Imposing the Albin's conditions on $X$, in this paper we obtain an exact asymptotic expansion of this probability as $u$ tends to infinity. Further, we establish the tail asymptotics of the supremum of a generalized skew-Gaussian process and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.

math.PR↗

arXiv2014

Finite-time ruin probability of aggregate Gaussian processes

Let $\left\{\sum_{i=1}^n λ_i X_i(t), t\in [0,T]\right\}$ be an aggregate Gaussian risk process with $X_i, i\leq n$ independent Gaussian processes satisfying Piterbarg conditions and $λ_i$'s given positive weights. In this paper we derive exact asymptotics of the finite-time ruin probability given by $$\mathbb{P}\left(\sup_{t\in[0,T]}\left(\sum_{i=1}^n λ_i X_i(t)- g(t) \right)>u\right)$$ as $u\to\infty$ for some general trend function $g$. Further, we derive asymptotic results for the finite-time ruin probabilities of risk processes perturbed by an aggregate Gaussian process.

math.PR↗

arXiv2013

Gaussian risk models with financial constraints

In this paper we investigate Gaussian risk models which include financial elements such as inflation and interest rates. For some general models for inflation and interest rates, we obtain an asymptotic expansion of the finite-time ruin probability for Gaussian risk models. Furthermore, we derive an approximation of the conditional ruin time by an exponential random variable as the initial capital tends to infinity.

math.PR↗

arXiv2013

Gaussian Approximation of Perturbed Chi-Square Risks

In this paper we show that the conditional distribution of perturbed chi-quare risks can be approximated by certain distributions including the Gaussian ones. Our results are of interest for conditional extreme value models and multivariate extremes as shown in three applications.

math.PR↗

arXiv2011

Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. The setting considered is that of a centered Gaussian process $X\equiv\{X(t):t\in\mathbb R\}$ with stationary increments and variance function $σ^2_X(\cdot)$, equipped with a deterministic drift $c>0$, reflected at 0: \[Q_X^{(c)}(t)=\sup_{-\infty<s\le t}(X(t)-X(s)-c(t-s)).\] We study the resulting stationary workload process $Q^{(c)}_X\equiv\{Q_X^{(c)}(t):t\ge0\}$ in the limiting regimes $c\to 0$ (heavy traffic) and $c\to\infty$ (light traffic). The primary contribution is that we show for both limiting regimes that, under mild regularity conditions on the variance function, there exists a normalizing function $δ(c)$ such that $Q^{(c)}_X(δ(c)\cdot)/σ_X(δ(c))$ converges to a non-trivial limit in $C[0,\infty)$.

math.PR↗

arXiv2011

On the infimum attained by a reflected Lévy process

This paper considers a Lévy-driven queue (i.e., a Lévy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided Lévy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of $\prob{M(T_u)> u}$ (for different classes of functions $T_u$ and $u$ large); here we have to distinguish between heavy-tailed and light-tailed scenarios.

math.PR↗