Extrema of multi-dimensional Gaussian processes over random intervals

Let $X(t),t\geq 0$ be an almost surely (a.s.) continuous centered Gaussian process with stationary increments and $X(0)=0$. Motivated by its applications to the hybrid fluid and ruin models, the seminal paper [1] derived the exact tail asymptotics of

with $\mathcal{T}$ being an independent of $X$ regularly varying random variable. Since then the study of the tail asymptotics of supremum on random interval has attracted substantial interest in the literature. We refer to [2, 3, 4, 5, 6, 7] for various extensions to general (non-centered) Gaussian or Gaussian-related processes. In the aforementioned contributions, various different tail distributions for $\mathcal{T}$ have been discussed, and it has been shown that the variability of $\mathcal{T}$ influences the form of the asymptotics of (1), leading to qualitatively different structures.

The primary aim of this paper is to analyze the asymptotics of a multi-dimensional counterpart of (1). More precisely, consider a multi-dimensional centered Gaussian process

with independent coordinates, each $X_{i}(t),t\geq 0,$ has stationary increments, a.s. continuous sample paths and $X_{i}(0)=0$, and let $\boldsymbol{\mathcal{T}}=(\mathcal{T}_{1},\cdots,\mathcal{T}_{n})$ be a regularly varying random vector with positive components, which is independent of $\boldsymbol{X}$. We are interested in the exact asymptotics of

where $c_{i}\in\mathbb{R}$, $a_{i}>0$, $i=1,2,\cdots,n$.

Extremal analysis of multi-dimensional Gaussian processes has been an active research area in recent years; see [8, 9, 10, 11, 12] and references therein. In most of these contributions, the asymptotic behaviour of the probability that $\boldsymbol{X}$ (possibly with trend) enters an upper orthant over a finite-time or infinite-time interval is discussed, this problem is also connected with the conjunction problem for Gaussian processes firstly studied by Worsley and Friston [13]. Investigations on the joint tail asymptotics of multiple extrema as defined in (3) have been known to be more challenging. Current literature has only focused on the case with deterministic times $\mathcal{T}_{1}=\cdots=\mathcal{T}_{n}$ and some additional assumptions on the correlation structure of $X_{i}$’s. In [14, 8] large deviation type results are obtained, and more recently in [15, 16] exact asymptotics are obtained for correlated two-dimensional Brownian motion. It is worth mentioning that a large derivation result for the multivariate maxima of a discrete Gaussian model is discussed recently in [17].

In order to avoid more technical difficulties, the coordinates of the multi-dimensional process $\boldsymbol{X}$ in (2) are assumed to be independent. The dependence among the extrema in (3) is driven by the structure of the multivariate regularly varying $\boldsymbol{\mathcal{T}}$. Interestingly, we observe in Theorem 3.1 that the form of the asymptotics of (3) is determined by the signs of the drifts $c_{i}$’s.

Apart from its theoretical interest, the motivation to analyse the asymptotic properties of $P(u)$ is related to numerous applications in modern multi-dimensional risk theory, financial mathematics or fluid queueing networks. For example, we consider an insurance company which runs $n$ lines of business. The surplus process of the $i$th business line can be modelled by a time-changed Gaussian process

where $a_{i}u>0$ is the initial capital (considered as a proportion of $u$ allocated to the $i$th business line, with $\sum_{i=1}^{n}a_{i}=1$), $c_{i}>0$ is the net premium rate, $X_{i}(t),t\geq 0$ is the net loss process, and $Y_{i}(t),t\geq 0$ is a positive increasing function modelling the so-called “operational time” for the $i$th business line. We refer to [18, 19] and [5] for detailed discussions on multi-dimensional risk models and time-changed risk models, respectively. Of interest in risk theory is the study of the probability of ruin of all the business lines within some finite (deterministic) time $T>0$, defined by

If additionally all the operational time processes $Y_{i}(t),t\geq 0$ have a.s. continuous sample paths, then we have $\varphi(u)=P(u)$ with $\boldsymbol{\mathcal{T}}=\boldsymbol{Y}(T)$, and thus the derived result can be applied to estimate this ruin probability. Note that the dependence among different business lines is introduced by the dependence among the operational time processes $Y_{i}$’s. As a simple example we can consider $Y_{i}(t)=\Theta_{i}t,$ $t\geq 0$, with $\boldsymbol{\Theta}=(\Theta_{1},\cdots,\Theta_{n})$ being a multivariate regularly varying random vector. Additionally, multi-dimensional time-changed (or subordinate) Gaussian processes have been recently proved to be good candidates for modelling the log-return processes of multiple assets; see, e.g., [20, 21, 22]. As the joint distribution of extrema of asset returns is important in finance problems, e.g., [23], we expect the obtained results for (3) might also be interesting in financial mathematics.

As a relevent application, we shall discuss a multi-dimensional regenerative model, which is motivated from its relevance to risk model and fluid queueing model. Essentially, the multi-dimensional regenerative process is a process with a random alternating environment, where an independent multi-dimensional fractional Brownian motion (fBm) with trend is assigned at each environment alternating time. We refer to Section 4 for more detail. By analysing a related multi-dimensional perturbed random walk, we obtain in Theorem 4.1 the ruin probability of the multi-dimensional regenerative model. This generalizes some of the results in [24] and [25] to the multi-dimensional setting. Note in passing that some related stochastic models with random sampling or resetting have been discussed in the recent literature; see, e.g., [26, 27, 28].

Organization of the rest of the paper: In Section 2 we introduce some notation, recall the definition of the multivariate regularly variation, and present some preliminary results on the extremes of one-dimensional Gaussian process. The result for (3) is displayed in Section 3, and the ruin probability of the multi-dimensional regenerative model is discussed in Section 4. The proofs are relegated to Section 5 and Section 6. Some useful results on multivariate regularly variation are discussed in the Appendix.

We shall use some standard notation which is common when dealing with vectors. All the operations on vectors are meant componentwise. For instance, for any given $\boldsymbol{x}=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ and $\boldsymbol{y}=(y_{1},\ldots,y_{n})\in\mathbb{R}^{n}$, we write $\boldsymbol{x}\boldsymbol{y}=(x_{1}y_{1},\cdots,x_{n}y_{n})$, and write $\boldsymbol{x}>\boldsymbol{y}$ if and only if $x_{i}>y_{i}$ for all $1\leq i\leq n$. Furthermore, for two positive functions $f,h$ and some $u_{0}\in[-\infty,\infty]$, write $f(u)\lesssim h(u)$ or $h(u)\gtrsim f(u)$ if $\limsup_{u\rightarrow u_{0}}f(u)/h(u)\leq 1$, write $h(u)\sim f(u)$ if $\lim_{u\rightarrow u_{0}}f(u)/h(u)=1$, write $f(u)=o(h(u))$ if $\lim_{u\rightarrow u_{0}}{f(u)}/{h(u)}=0$, and write $f(u)\asymp h(u)$ if $f(u)/h(u)$ is bounded from both below and above for all sufficiently large $u$.

Next, let us recall the definition and some implications of multivariate regularly variation. We refer to [29, 30, 31] for more detailed discussions. Let $\overline{\mathbb{R}}_{0}^{n}=\overline{\mathbb{R}}^{n}\setminus\{\boldsymbol{0}\}$ with $\overline{\mathbb{R}}=\mathbb{R}\cup\{-\infty,\infty\}$. An $\mathbb{R}^{n}$-valued random vector $\boldsymbol{X}$ is said to be regularly varying if there exists a non-null Radon measure $\nu$ on the Borel $\sigma$-field $\mathcal{B}(\overline{\mathbb{R}}_{0}^{n})$ with $\nu(\overline{\mathbb{R}}^{n}\setminus\mathbb{R}^{n})=0$ such that

Here $\left\lvert\ \cdot\ \right\rvert$ is any norm in $\mathbb{R}^{n}$ and $\overset{v}{\rightarrow}$ refers to vague convergence on $\mathcal{B}(\overline{\mathbb{R}}_{0}^{n})$. It is known that $\nu$ necessarily satisfies the homogeneous property $\nu(sK)=s^{-\alpha}\nu(K)$, $s>0$, for some $\alpha>0$ and all Borel set $K$ in $\mathcal{B}(\overline{\mathbb{R}}_{0}^{n})$. In what follows, we say that such defined $\boldsymbol{X}$ is regularly varying with index $\alpha$ and limiting measure $\nu$. An implication of the homogeneity property of $\nu$ is that all the rectangle sets of the form $[\boldsymbol{a},\boldsymbol{b}]=\{\boldsymbol{x}:\boldsymbol{a}\leq\boldsymbol{x}\leq\boldsymbol{b}\}$ in $\overline{\mathbb{R}}_{0}^{n}$ are $\nu$-continuity sets. Furthermore, we have that $\left\lvert\boldsymbol{X}\right\rvert$ is regularly varying at infinity with index $\alpha$, i.e., $\mathbb{P}\left\{\left\lvert\boldsymbol{X}\right\rvert>x\right\}\sim x^{-\alpha}L(x),x\rightarrow\infty$, with some slowly varying function $L(x)$. Some useful results on the multivariate regularly variation are discussed in Appendix.

In what follows, we review some results on the extremes of one-dimensional Gaussian process with nagetive drift derived in [32]. Let $X(t),t\geq 0$ be an a.s. continuous centered Gaussian process with stationary increments and $X(0)=0$, and let $c>0$ be some constant. We shall present the exact asymptotics of

Below are some assumptions that the variance function $\sigma^{2}(t)=\mathrm{Var}(X(t))$ might satisfy:

• C1:

  $\sigma$ is continuous on $[0,\infty)$ and ultimately strictly increasing;

• C2:

  $\sigma$ is regularly varying at infinity with index $H$ for some $H\in(0,1)$;

• C3:

  $\sigma$ is regularly varying at 0 with index $\lambda$ for some $\lambda\in(0,1)$;

• C4:

  $\sigma^{2}$ is ultimately twice continuously differentiable and its first derivative $\dot{\sigma}^{2}$ and second derivative $\ddot{\sigma}^{2}$ are both ultimately monotone.

Note that in the above $\dot{\sigma}^{2}$ and $\ddot{\sigma}^{2}$ denote the first and second derivative of $\sigma^{2}$, not the square of the derivatives of $\sigma$. In the sequel, provided it exists we denote by $\overleftarrow{\sigma}$ an asymptotic inverse near infinity or zero of $\sigma$; recall that it is (asymptotically uniquely) defined by $\overleftarrow{\sigma}(\sigma(t))\sim\sigma(\overleftarrow{\sigma}(t))\sim t.$ It depends on the context whether $\overleftarrow{\sigma}$ is an asymptotic inverse near zero or infinity.

One known example that satisfies the assumptions C1-C4 is the fBm $\{B_{H}(t),t\geq 0\}$ with Hurst index $H\in(0,1)$, i.e., an $H$-self-similar centered Gaussian process with stationary increments and covariance function given by

We introduce the following notation:

For an a.s. continuous centered Gaussian process $Z(t),t\geq 0$ with stationary increments and variance function $\sigma_{Z}^{2}$, we define the generalized Pickands constant

provided both the expectation and the limit exist. When $Z=B_{H}$ the constant $\mathcal{H}_{B_{H}}$ is the well-known Pickands constant; see [33]. For convenience, sometimes we also write $\mathcal{H}_{\sigma_{Z}^{2}}$ for $\mathcal{H}_{Z}$. Denote in the following by $\Psi(\cdot)$ the survival function of the $N(0,1)$ distribution. It is known that

As a special case of the Proposition 2.1 we have the following result (see Corollary 1 in [32] or [19]). This will be useful in the proofs below.

Without loss of generality, we assume that in (3) there are $n_{-}$ coordinates with negative drift, $n_{0}$ coordinates without drift and $n_{+}$ coordinates with positive drift, i.e.,

where $0\leq n_{-},n_{0},n_{+}\leq n$ such that $n_{-}+n_{0}+n_{+}=n$. We impose the following assumptions for the standard deviation functions $\sigma_{i}(t)=\sqrt{Var(X_{i}(t))}$ of the Gaussian processes $X_{i}(t),i=1,\cdots,n$.

Assumption I: For $i=1,\cdots,n_{-},$ $\sigma_{i}(t)$ satisfies the assumptions C1-C4 with the parameters involved indexed by $i$. For $i=n_{-}+1,\cdots,n_{-}+n_{0},$ $\sigma_{i}(t)$ satisfies the assumptions C1-C3 with the parameters involved indexed by $i$. For $i=n_{-}+n_{0}+1,\cdots,n,$ $\sigma_{i}(t)$ satisfies the assumptions C1-C2 with the parameters involved indexed by $i$.

Denote

Given a Radon measure $\nu$, define

where $\boldsymbol{\xi}^{-1/{\boldsymbol{H}}}K=\{(\xi_{1}^{-1/{H_{1}}}d_{1},\cdots,\xi_{n}^{-1/{H_{n}}}d_{n}),(d_{1},\cdots,d_{n})\in K\}$. Further, note that for $i=1,\cdots,n_{-},$ (where $c_{i}<0$), the asymptotic formula, as $u\rightarrow\infty$, of

is available from Proposition 2.1 under Assumption I.

As it is known in the literature that the maximum of random processes over random interval is relevant to the regenerated models (e.g., [24, 25]), this section is focused on a multi-dimensional regenerative model which is motivated from its applications in queueing theory and ruin theory. More precisely, there are four elements in this model: Two sequences of strictly positive random variables, $\{T_{i}:i\geq 1\}$ and $\{S_{i}:i\geq 1\}$, and two sequences of $n$-dimensional processes, $\{\{\boldsymbol{X}^{(i)}(t),t\geq 0\}:i\geq 1\}$ and $\{\{\boldsymbol{Y}^{(i)}(t),t\geq 0\}:i\geq 1\}$, where $\boldsymbol{X}^{(i)}(t)=(X_{1}^{(i)}(t),\cdots,X_{n}^{(i)}(t))$ and $\boldsymbol{Y}^{(i)}(t)=(Y_{1}^{(i)}(t),\cdots,Y_{n}^{(i)}(t))$. We assume that the above four elements are mutually independent. Here $T_{i},S_{i}$ are two successive times representing the random length of the alternating environment (called $T$-stage and $S$-stage), and we assume a $T$-stage starts at time 0. The model grows according to $\{\boldsymbol{X}^{(i)}(t),t\geq 0\}$ during the $i$th $T$-stage and according to $\{\boldsymbol{Y}^{(i)}(t),t\geq 0\}$ during the $i$th $S$-stage.

Based on the above we define an alternating renewal process with renewal epochs

with $V_{i}=(T_{1}+S_{1})+\cdots+(T_{i}+S_{i})$ which is the $i$th environment cycle time. Then the resulting $n$-dimensional process $\boldsymbol{Z}(t)=(Z_{1}(t),\cdots,Z_{n}(t))$, is defined as

Note that this is a multi-dimensional regenerative process with regeneration epochs $V_{i}$. This is a generalization of the one-dimensional model discussed in [26].

We assume that $\{\{\boldsymbol{X}^{(i)}(t),t\geq 0\}:i\geq 1\}$ and $\{\{\boldsymbol{Y}^{(i)}(t),t\geq 0\}:i\geq 1\}$ are independent samples of $\{\boldsymbol{X}(t),t\geq 0\}$ and $\{\boldsymbol{Y}(t),t\geq 0\}$, respectively, where

with all the fBm’s $B_{H_{j}},\widetilde{B}_{\widetilde{H}_{j}}$ being mutually independent and $p_{j},q_{j}>0,1\leq j\leq n$. Suppose that $(T_{i},S_{i}),i\geq 1$ are independent samples of $(T,S)$ and $T$ is regularly varying with index $\lambda>1$. We further assume that

For notational simplicity we shall restrict ourselves to the 2-dimensional case. The general $n$-dimensional problem can be analysed similarly. Thus, for the rest of this section and related proofs in Section 6, all vectors (or multi-dimensional processes) are considered to be two-dimensional ones.

We are interested in the asymptotics of the following tail probability

with $a_{1},a_{2}>0$. In the fluid queueing context, $Q(u)$ can be interpreted as the probability that both buffers overflow in some environment cycle. In the insurance context, $Q(u)$ can be interpreted as the probability that in some business cycle the two lines of business of the insurer are both ruined (not necessarily at the same time). Similar one-dimensional models have been discussed in the literature; see, e.g., [25, 24, 18].

We introduce the following notation:

Then we have

Note that $\boldsymbol{U}^{(n)},n\geq 0$ and $\boldsymbol{M}^{(n)},n\geq 0$ are both IID sequences. By the second assumption in (12) we have

which ensures that the event in the above probability is a rare event for large $u$, i.e., $Q(u)\rightarrow 0,$ as $u\rightarrow\infty$.

It is noted that our question now becomes an exit problem of a 2-dimensional perturbed random walk. The exit problems of multi-dimensional random walk has been discussed in many papers, e.g., [31]. However, it seems that multi-dimensional perturbed random walk has not been discussed in the existing literature.

This section is devoted to the proof of Theorem 3.1 followed by a short proof of Example 3.3.