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This paper proposes three types of Bayesian CART (or BCART) models for aggregate claim amount, namely, frequency-severity models, sequential models and joint models. We propose a general framework for the BCART models applicable to data with multivariate responses, which is particularly useful for the joint BCART models with a bivariate response: the number of claims and aggregate claim amount. To facilitate frequency-severity modeling, we investigate BCART models for the right-skewed and heavy-tailed claim severity data by using various distributions. We discover that the Weibull distribution is superior to gamma and lognormal distributions, due to its ability to capture different tail characteristics in tree models. Additionally, we find that sequential BCART models and joint BCART models, which incorporate dependence between the number of claims and average severity, are beneficial and thus preferable to the frequency-severity BCART models in which independence is assumed. The effectiveness of these models' performance is illustrated by carefully designed simulations and real insurance data.

Accuracy and interpretability of a (non-life) insurance pricing model are essential qualities to ensure fair and transparent premiums for policy-holders, that reflect their risk. In recent years, the classification and regression trees (CARTs) and their ensembles have gained popularity in the actuarial literature, since they offer good prediction performance and are relatively easily interpretable. In this paper, we introduce Bayesian CART models for insurance pricing, with a particular focus on claims frequency modelling. Additionally to the common Poisson and negative binomial (NB) distributions used for claims frequency, we implement Bayesian CART for the zero-inflated Poisson (ZIP) distribution to address the difficulty arising from the imbalanced insurance claims data. To this end, we introduce a general MCMC algorithm using data augmentation methods for posterior tree exploration. We also introduce the deviance information criterion (DIC) for the tree model selection. The proposed models are able to identify trees which can better classify the policy-holders into risk groups. Some simulations and real insurance data will be discussed to illustrate the applicability of these models., Comment: 46 pages

In this paper, we study the asymptotic distribution of the maxima of suprema of dependent Gaussian processes with trend. For different scales of the time horizon we obtain different normalizing functions for the convergence of the maxima. The obtained results not only have potential applications in estimating the delay of certain Gaussian fork-join queueing systems but also provide interesting insights to the extreme value theory for triangular arrays of random variables with row-wise dependence., Comment: 21 pages

We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised $k$th order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random variable, a Normal random variable or a mixture of them, according to three conditions deduced through the model parameters. Remarkably, this phenomenon resembles that for the stationary Normal sequence. We also show that various moments of the normalised $k$th order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyze various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory., Comment: 29 pages

This paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as $$ P(u):=\mathbb{P}\left\{\cap_{i=1}^n \left(\sup_{t\in[0,\mathcal{T}_i]} ( X_{i}(t) +c_i t )>a_i u \right)\right\}, \ \ \ u\to\infty, $$ where $X_i(t), t\ge0$, $i=1,2,\cdots,n,$ are independent centered Gaussian processes with stationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \cdots, \mathcal{T}_n)$ is a regularly varying random vector with positive components, which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$, $a_i>0$, $i=1,2,\cdots,n$. Our result shows that the structure of the asymptotics of $P(u)$ is determined by the signs of the drifts $c_i$'s. We also discuss a relevant multi-dimensional regenerative model and derive the corresponding ruin probability.

We derive the exact asymptotics of \[ P\left( \sup_{t\ge 0} \Bigl( X_1(t) - \mu_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - \mu_2 s\Bigr)> u \right), \ \ u\to\infty, \] where $(X_1(t),X_2(s))_{t,s\ge0}$ is a correlated two-dimensional Brownian motion with correlation $\rho\in[-1,1]$ and $\mu_1,\mu_2>0$. It appears that the play between $\rho$ and $\mu_1,\mu_2$ leads to several types of asymptotics. Although the exponent in the asymptotics as a function of $\rho$ is continuous, one can observe different types of prefactor functions depending on the range of $\rho$, which constitute a phase-type transition phenomena.

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 $\rho$ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem., Comment: 20

Consider a multi-dimensional Brownian motion which models the surplus processes of multiple lines of business of an insurance company. Our main result gives exact asymptotics for the cumulative Parisian ruin probability as the initial capital tends to infinity. An asymptotic distribution for the conditional cumulative Parisian ruin time is also derived. The obtained results on the cumulative Parisian ruin can be seen as generalizations of some of the results derived in Debicki et al (2018, Stochastic Processes and Their Applications). As a particular interesting case, the two-dimensional Brownian motion risk model is discussed in detail., Comment: 22 pages

In this paper, we analyze a multivariate counterpart of the generalized weighted Kolmogorov-Smirnov statistic, which is the supremum of weighted locally stationary chi-square process over non-compact interval. The boundedness and the exact tail asymptotic behavior of the statistics are derived. We illustrate our findings by two examples where the statistic is defined by Brownian bridge and fractional Brownian motion respectively., Comment: 16 pages

This paper derives an exact asymptotic expression for \[ \mathbb{P}_{\mathbf{x}_u}\{\exists_{t\ge0} \mathbf{X}(t)- \boldsymbol{\mu}t\in \mathcal{U} \}, \ \ {\rm as}\ \ u\to\infty, \] where $\mathbf{X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated $d$-dimensional Brownian motion starting at the point $\mathbf{x}_u=-\boldsymbol{\alpha}u$ with $\boldsymbol{\alpha}\in \mathbb{R}^d$, $\boldsymbol{\mu} \in \mathbb{R}^d$ and $\mathcal{U}=\prod_{i=1}^d [0,\infty)$. The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to $\mathcal{U}$, which depends on the dimension-reduction phenomena., Comment: 32 pages

In this short note, we derive explicit formulas for the joint densities of the time to ruin and the number of claims until ruin in perturbed classical risk models, by constructing several auxiliary random processes., Comment: 9

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$ P\left(\exists_{t \in [0,T]} \forall_{i=1 ... n} X_i(t)> u \right) $$ as $u\to\infty$, for both locally stationary $X_i$'s and $X_i$'s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands-Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes., Comment: 20 pages

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., Comment: 20

Chi-square processes with trend appear naturally as limiting processes in various statistical models. In this paper we are concerned with the exact tail asymptotics of the supremum taken over (0; 1) of a class of locally stationary chi-square processes with particular admissible trends. An important tool for establishing our results is a weak version of Slepian's lemma for chi-square processes. Some special cases including squared Brownian bridge and Bessel process are discussed., Comment: 26 pages in Stochastic Processes and their Applications, 2016

This paper studies the supremum of a chi-square process with trend over a threshold-dependent-time horizon. Under the assumption that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square process with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed., Comment: 23 pages

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times., Comment: 14 pages

This contribution establishes exact tail asymptotics of $\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous Gaussian random fields $X$ on a bounded convex set $\mathbf{E}\subset\mathbb{R}^2$, with variance function that attains its maximum on a segment on $\mathbf{E}$. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes., Comment: Published at http://dx.doi.org/10.1214/14-AOP994 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Let $\left\{\sum_{i=1}^n \lambda_i X_i(t), t\in [0,T]\right\}$ be an aggregate Gaussian risk process with $X_i, i\leq n$ independent Gaussian processes satisfying Piterbarg conditions and $\lambda_i$'s given positive weights. In this paper we derive exact asymptotics of the finite-time ruin probability given by $$\mathbb{P}\left(\sup_{t\in[0,T]}\left(\sum_{i=1}^n \lambda_i X_i(t)- g(t) \right)>u\right)$$ 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., Comment: 11 pages

Random shifting typically appears in credibility models whereas random scaling is often encountered in stochastic models for claim sizes reflecting the time-value property of money. In this article we discuss some aspects of random shifting and random scaling in insurance focusing in particular on credibility models, dependence structure of claim sizes in collective risk models, and extreme value models for the joint dependence of large losses. We show that specifying certain actuarial models using random shifting or scaling has some advantages for both theoretical treatments and practical applications., Comment: 11 pages

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., Comment: 20 pages, revised version

Define a $\gamma$-reflected process $W_\gamma(t)=Y_H(t)-\gamma\inf_{s\in[0,t]}Y_H(s)$, $t\ge0$ with input process $\{Y_H(t), t\ge 0\}$ which is a fractional Brownian motion with Hurst index $H\in (0,1)$ and a negative linear trend. In risk theory $R_\gamma(t)=u-W_\gamma(t), t\ge0$ is referred to as the risk process with tax of a loss-carry-forward type, whereas in queueing theory $W_1$ is referred to as the queue length process. In this paper, we investigate the ruin probability and the ruin time of the risk process $R_\gamma, \gamma \in [0,1]$ over a surplus dependent time interval $[0, T_u]$.

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants $u,T$, define the set of conjunctions $C_{[0,T],u}:=\{t\in [0,T]: \min_{1 \le i \le n} X_i(t) \ge u\}.$ Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of $ P(C_{[0,T],u} \not=\varphi)$ as $u\to\infty$. Moreover, we establish the Berman sojourn limit theorem for the random process $\{\min_{1 \le i \le n} X_i(t), t\ge0\}$ and derive the tail asymptotics of the supremum of each order statistics process., Comment: 11 pages, Theorem 2.3 is new in this version

Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ an independent of $X$ non-negative random variable. As an application, we derive the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes., Comment: 15 pages

Define a gamma-reflected process W_\gamma(t)=Y_H(t)-\gamma\inf_{s\in[0,t]}Y_H(s), t\ge0 with input process {Y_H(t), t\ge 0} which is a fractional Brownian motion with Hurst index H\in (0,1) and a negative linear trend. In risk theory {u-W_\gamma(t), t\ge 0} is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserve $u$ goes to infinity. In this paper we show that for the gamma-reflected process the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of non-homogeneous Gaussian random fields defined by Y_H which are also investigated in this contribution., Comment: 15 pages

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., Comment: To appear in Scandinavian Actuarial Journal

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.

Let $\{X_i(t),t\ge0\}, i=1,2$ be two standard fractional Brownian motions being jointly Gaussian with constant cross-correlation. In this paper we derive the exact asymptotics of the joint survival function $$ \mathbb{P}\{\sup_{s\in[0,1]}X_1(s)>u,\ \sup_{t\in[0,1]}X_2(t)>u\} $$ as $u\rightarrow \infty$. A novel finding of this contribution is the exponential approximation of the joint conditional first passage times of $X_1, X_2$. As a by-product we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields., Comment: 16 pages, title changed

Let $\chi_n(t) = (\sum_{i=1}^n X_i^2(t))^{1/2},t\ge0$ be a chi-process with $n$ degrees of freedom where $X_i$'s are independent copies of some generic centered Gaussian process $X$. This paper derives the exact asymptotic behavior of P{\sup_{t\in[0,T]} \chi_n(t)>u} as u \to \infty, where $T$ is a given positive constant, and $g(\cdot)$ is some non-negative bounded measurable function. The case $g(t)\equiv0$ is investigated in numerous contributions by V.I. Piterbarg. Our novel asymptotic results for both stationary and non-stationary $X$are referred to as Piterbarg theorems for chi-processes with trend., Comment: 22 pages

This contribution derives the exact asymptotic behaviour of the supremum of alpha(t)-locally stationary Gaussian random fields over a finite hypercube. We present two applications of our result; the first one deals with extremes of ggregate multifractional Brownian motions, whereas the second application establishes the exact asymptotics of the supremum of chi-processes generated by multifractional Brownian motions., Comment: 22 pages

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 $c>0,\gamma \in [0,1]$ two given constants. In this paper we establish the exact tail asymptotic behaviour of $\sup_{t\in [0,T]} W_\gamma(t)$ 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., Comment: 15 pages