The time of ultimate recovery in Gaussian risk model

Abstract

We analyze the distance $\mathcal {R}_{T}(u)$ between the first and the last passage time of {X(t) − ct : t ∈ [0, T]} at level u in time horizon T ∈ (0, ∞], 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

$$\lim\limits_{u\to\infty}\mathbb{P} \left( \mathcal{R}_{T}(u)>{\Delta}(u)x \right)=G(x), $$

for x ≥ 0. We distinguish two scenarios: T < ∞ and T = ∞, that lead to qualitatively different asymptotics. The obtained results provide exact asymptotics of the ultimate recovery time after the ruin in Gaussian risk model.

Appendix

In this section we give a variant of Theorem 2.1 in Dȩbicki et al. (2017a). Let $\xi _{u,\tau _{u}}$ be a family of Gaussian random fields given by

$$ \xi_{u,\tau_u}(t)=\frac{Z_{u,\tau_u}(t)}{1+h_{u,\tau_u}(t)}, \quad t\in [0,S], \tau_u\in K_u, $$

(38)

where $Z_{u,\tau _{u}}$ is a family of centered Gaussian random fields with continuous trajectories and unit variance, $h_{u,\tau _{u}}\in C([0,S])$, S > 0 and K_u is a set of index. We investigate the asymptotics of

$$\mathbb{P} \left( {\Gamma}^{\prime}(x,S;\xi_{u,\tau_{u}})>g_{u,\tau_{u}}, D_{u,\tau_{u}} \right) $$

as u→∞ where $g_{u,\tau _{u}}$ is a series of positive functions of u,

$$\begin{array}{@{}rcl@{}} D_{u,\tau_{u}}=\{Z_{u,\tau_{u}}(0)=g_{u,\tau_{u}}-\frac{w}{g_{u,\tau_{u}}}, w\in D\} \end{array} $$

(39)

with D = (0, ∞), (−∞, 0) or $D=\mathbb {R}$ and ${\Gamma }^{\prime }: C([0,S]){\rightarrow } \mathbb {R}$, 0 ≤ x ≤ S is a real-valued continuous functional defined by

$$ {\Gamma}^{\prime}(x,S; f)=\sup\limits_{t\in [x,S]}\min\left( f(t), \sup_{s\in [0,t-x]}f(s)\right), \quad f\in C([0,S]). $$

(40)

In order to avoid trivialities, we assume that

$$\lim\limits_{u{\rightarrow}{\infty}}\mathbb{P} \left( {\Gamma}^{\prime}(x,S;\xi_{u,\tau_{u}})>g_{u,\tau_{u}}, D_{u,\tau_{u}} \right)= 0 $$

and observe that $\mathbb {P} \left ({\Gamma }^{\prime }(x,S;\xi _{u,\tau _{u}})>g_{u,\tau _{u}}, D_{u,\tau _{u}} \right )=\mathbb {P} \left ({\Gamma }^{\prime }(x,S;\xi _{u,\tau _{u}})>g_{u,\tau _{u}} \right )$ if $D={\mathbb {R}}$.

As in Dȩbicki et al. (2017a) (see Theorem 2.1), we impose the following assumptions:

D0::

$\lim _{u{\rightarrow }{\infty }}\inf _{\tau _{u}\in K_{u}}g_{u,\tau _{u}}={\infty }$.

D1::

There exist ρ(t), regularly varying function at 0 with index 2α₀ ∈ (0, 2] and b_i(u) > 0, i = 1, 2 satisfying $\lim _{u{\rightarrow }{\infty }}b_{i}(u)= 0, i = 1,2$ and $\lim _{u{\rightarrow }{\infty }}\frac {b_{1}(u)}{b_{2}(u)}=\nu \in [0,{\infty })$ such that

$$\lim\limits_{u{\rightarrow}{\infty}}\sup\limits_{{\tau_u} \in K_u}\sup\limits_{s,t\in [0,S], s\neq t}\left| \left( g_{u,\tau_u}\right)^{2}\frac{1-Corr(Z_{u,\tau_u}(t), Z_{u,\tau_u}(s))}{\frac{\rho(b_1(u)|t-s|)}{\rho(b_2(u))}}-1\right|= 0. $$

D2::

There exists h ∈ C([0, S]) such that

$$\lim\limits_{u{\rightarrow}{\infty}}\sup\limits_{\tau_{u}\in K_{u}}\sup\limits_{t\in [0,S]}\left|(g_{u,\tau_{u}})^{2}h_{u,\tau_{u}}(t)-h(t)\right|= 0. $$

Lemma 3.5

Let$\xi _{u,\tau _{u}}$be definedas in Eq. 38and Γ^′be defined in Eq. 40. Assume thatD0-D2are satisfied. Then, forD_u, τ_udefined in Eq. 39withD = (0, ∞), (−∞, 0) or$D=\mathbb {R}$,

$$\lim\limits_{u{\rightarrow}{\infty}}\sup\limits_{\tau_{u}\in K_{u}}\left| \frac{\mathbb{P} \left( {\Gamma}^{\prime}(x,S;\xi_{u,\tau_{u}})>g_{u,\tau_{u}}, D_{u,\tau_{u}} \right)}{{\Psi}\left( g_{u,\tau_{u}}\right)} - {\int}_{D} e^{w} \mathbb{P} \left( {\Gamma}^{\prime}(x,S;\!\nu^{\alpha_{0}} B_{\alpha_{0}}(t) - \nu^{2\alpha_{0}}|t|^{2\alpha_{0}} - h(t))\!>\!w \right)dw\right| = 0. $$

Proof

Conditioning on the event that $Z_{u,\tau _{u}}(0)=g_{u,\tau _{u}}-\frac {w}{g_{u,\tau _{u}}}$ and noting that $D_{u,\tau _{u}}=\{Z_{u,\tau _{u}}(0)=g_{u,\tau _{u}}-\frac {w}{g_{u,\tau _{u}}}, w\in D\}$, we have

$$\begin{array}{@{}rcl@{}} &&\mathbb{P} \left( {\Gamma}^{\prime}(x,S;\xi_{u,\tau_u})>g_{u,\tau_u}, D_{u,\tau_u} \right)\\ &&\quad = \frac{1}{\sqrt{2\pi}g_{u,\tau_u}}{\int}_{\mathbb{R}}e^{-\frac{\left( g_{u,\tau_u}-\frac{w}{g_{u,\tau_u}}\right)^{2}}{2}}\mathbb{P} \left( {\Gamma}^{\prime}(x,S;\xi_{u,\tau_u})>g_{u,\tau_u}, D_{u,\tau_u}\left|Z_{u,\tau_u}(0)=g_{u,\tau_u}-\frac{w}{g_{u,\tau_u}} \right.\right)dw\\ && \quad = \frac{e^{-\frac{(g_{u,\tau_u})^{2}}{2}}}{\sqrt{2\pi}g_{u,\tau_u}}{\int}_{D}e^{w-\frac{w^{2}}{2(g_{u,\tau_u})^{2}}}\mathbb{P} \left( {\Gamma}^{\prime}(x,S;\xi_{u,\tau_u})>g_{u,\tau_u}\left|Z_{u,\tau_u}(0)=g_{u,\tau_u}-\frac{w}{g_{u,\tau_u}} \right.\right)dw. \end{array} $$

Using the same procedure as in the proof of Theorem 2.1 in Dȩbicki et al. (2017a), we can show that

$${\int}_{D}e^{w-\frac{w^{2}}{2(g_{u,\tau_{u}})^{2}}}\mathbb{P} \left( {\Gamma}^{\prime}(x,S;\xi_{u,\tau_{u}})>g_{u,\tau_{u}}\left|Z_{u,\tau_{u}}(0)=g_{u,\tau_{u}}-\frac{w}{g_{u,\tau_{u}}} \right.\right)dw $$

uniformly converges to

$${\int}_{D}e^{w}\mathbb{P} \left( {\Gamma}^{\prime}(x,S;\nu^{\alpha_{0}} B_{\alpha_{0}}(t)-\nu^{2\alpha_{0}}|t|^{2\alpha_{0}}-h(t))>w \right)dw $$

with respect to τ_u ∈ K_u. This completes the proof. □

Remark 3.6

Lemma 3.5 also holds if we substitute Γ^′ by _supt∈[x, S]f(t), with f ∈ C[x, S], x ≥ 0 and D, a measurable subset of $\mathbb {R}$ with positive Lebesgue measure.