The Speed of Convergence of the Threshold Estimator of Ruin Probability under the Tempered α-Stable Lévy Subordinator

In this paper, a nonparametric estimator of ruin probability is introduced in a spectrally negative Lévy process where the jump component is a tempered α-stable subordinator. Given a discrete record of high-frequency data, a threshold technique is proposed to estimate the mean of the jump size and use the Fourier transform and the Pollaczek–Khinchin formula to construct the estimator of ruin probability. The convergence rate of the integrated squared error for the estimator is studied.

In Asmussen and Albrecher [23], an analytic (or probabilistic) approach was suggested, and it needs much more detailed information about the risk model, such as the claim size distribution. However, in practical situations, it is not easy to obtain the specific distribution information. Instead, one observes the surplus process at some discrete time points. Then, a statistical methodology can be directly used to estimate the claim size distribution with the observed data. In Zhang and Yang [10], a nonparametric estimator of ruin probability was proposed, based on the Pollaczek-Khinchin formula and the Fourier transform in a pure-jump Lévy risk model. This estimation approach was extended by Zhang and Yang [11] to a spectrally negative Lévy risk model. Subsequently, Shimizu and Zhang [22] estimated the Gerber-Shiu function for an insurance surplus process driven by a Lévy subordinator. In Zhang and Yang [10] and Comte and Genon-Catalot [24], they considered high-frequency sampling with n discrete time observations of step width h n > 0 and derived asymptotics under the framework that h n → 0 and nh n → ∞.
In the present work, our interest is to estimate ruin probability for a spectrally negative Lévy risk model under the above framework. Assume that the surplus of the risk model can be observed at a sequence of discrete time points {t n k = kh n , k = 0, 1, 2, 3, ...} with h n ≥ 0 being the length of the sampling interval. Without observing the jump and diffusion parts of the risk model, it is challenging to estimate ruin probability, since it depends on both parts in a spectrally negative Lévy risk model. In Mancini [25][26][27][28] and Shimizu [29,30], they developed a threshold technique for identifying the times when jump sizes exceed a suitably defined threshold. Using the threshold technique and the Fourier transform, an estimator of ruin probability is constructed, and the convergence rate of its integrated squared error is obtained.
The remainder of this paper is organized as follows. In Section 2, the risk model, as well as some assumptions for the asymptotic theory are introduced. In Section 3, some estimators are suggested, based on the Fourier transform and the threshold technique. In Section 4, the convergence rate of our estimators is established. In Section 5, we conclude this paper.

Risk Model and Some Assumptions
A spectrally negative Lévy process is specified by: where c > 0 is a parameter, σ > 0 represents the perturbation coefficient, W t is a standard Brownian motion, and J t is a subordinator. Suppose that W t and J t are independent of each other. Then, the characteristic exponent of Y t is given by: where ν is the Lévy measure on (0, ∞). By Sato [31], it can be rewritten as: where L t is the sum of jumps over [0, t] with the jump size larger than one, and M * t is the sum of jumps over [0, t] with the jump size less than one. Specifically, L t = t 0 x>1 xµ(ds, dx) = ∑ N t k=1 γ k , where µ is the Poisson random measure of J t such that E[µ(ds, dx)] = ν(dx)ds, N t is a Poisson process, and γ 1 , γ 2 , γ 3 , ... are i.i.d. random variables, that is L t is a compound Poisson process representing the jumps of J t with the jump size larger than one. Process M * t admits decomposition, Suppose that γ k , N t , and M t are independent of each other.
Let u > 0 be the initial surplus of an insurance company. Then, the surplus at time t can be modeled by: where c is the rate of the premium, σ represents the perturbation coefficient, J t denotes the claim payments and other expenses in insurance businesses, and W t is a perturbation.

Ruin Probability and Its Fourier Transform
The infinite-time horizon ruin probability Φ(u) is defined as: By Equation (1) in Zhang and Yang [11], Φ(u) admits the following representation: x 0 ν(y, ∞)dy, and G is determined by the Fourier transform ∞ 0 e isx dG(x) = c{c − σ 2 2 is} −1 . Setting Φ(u) ≡ 0 for u < 0, Zhang and Yang [11] obtained the Fourier transform of Φ(u): Once an estimator of F Φ (s) is available, Φ(u) can be estimated by the inverse Fourier transform.

Estimation of Ruin Probability
Suppose that a discrete sample Y n = {Y t n i | t n i = ih n ; i = 0, 1, 2..., n} can be observed.
Our interest is to estimate Φ(u) by Z 1 , Z 2 , ..., Z n when Lévy measure ν and perturbation coefficient σ are unknown.
If one can estimate ρ and ψ Y (−s) in (6), then F Φ (s) can be estimated with the plug-in device. Inspired by Zhang and Yang [10,11] and You and Yin [32], we define the estimator of ψ Y (s):ψ To estimate ρ = µ 1 c , we need to estimate µ 1 , the mean of J 1 . Zhang and Yang [11] proposed to estimate µ 1 , by: . Ideally, we hope that the estimator of µ 1 is 1 nh n ∑ n k=1 (J t n k − J t n k−1 ), but we cannot observe a discrete sample J n = {J t n i | t n i = ih n ; i = 0, 1, 2..., n}. To this end, we introduce a threshold technique. Motivated by Shimizu [29,30] and Mancini [26,27], we introduce the filter: where r n > 0 is a suitable threshold parameter dependent on n such that lim h n →0 r n = 0. Let C n k := {ω ∈ Ω : (ch n − Z k ) ≤ r n } be the complement of D n k . By (9), if ch n − Z k > r n , we can detect the existence of a jump in an interval (t n k−1 , t n k ], and then, we take ch n − Z k as an approximation to J t n k − J t n k−1 . This leads to a natural estimate of µ 1 : Then, ρ is estimated by:ρ Combining (6), (7), and (11) leads to our estimator: Note that the above estimate has no definition at s = 0. When s → 0,ψ Y (−s) → 0, and thus,F Φ (s) may behave erratically. Applying the inverse Fourier transform and removing a small neighborhood of s = 0, we propose to estimate Φ(u) by: where m n and M n are positive threshold numbers such that m n → 0 and M n → ∞ as n → ∞.

Asymptotic Properties of Estimators
In this section, the asymptotic properties ofρ andΦ(u) are studied. For the ease of exposure, we first introduce some notations. For integer k = 1, 2, µ k := ∞ 0 x k ν(dx). For any two positive sequences {x n } ∞ n=1 and {y n } ∞ n=1 , x n y n means that x n ≤ Cy n for some constant C and large index n. For any function f (x) with support (0, ∞), define . Next, we make the following assumptions for our theoretical results: Assumption 1. The safety loading condition holds, i.e., c − µ 1 < ∞.
To establish the convergence rate ofΦ(u), we need to calibrate the estimation errors ofρ. The following Theorem 1 gives the rate of convergence ofρ. Theorem 1. Let r n = h θ n with θ ∈ (0, 1/2). Then, under Assumptions 1 and 2, where , and ∆ k N = (N t n k − N t n k−1 ). By (9) and (11), we have: where: In the following, we study each of I 1 to I 4 .
(18) By Mancini [25], {∆ k L − ∆ k W > h θ n } is equal to {∆ k N = 1} almost surely for small h n . Thus, for small h n , Rewrite: Then, due to the independence of γ k , N t , and W t , by the central limit theorem, we ). [32], we have:
By Mancini [27], we obtain that: (iii) Applying the central limit theorem, we obtain: where T h n D − → N (0, σ 2 M c 2 ). (iv) Now, let us consider the last term of (17).
By (A.29)-(A.32) in You and Yin [32], we have ). (24) Next, we show that the second term of (23) is of order O(r 1−α n ). In fact, Using Assumption 2 and the law of large numbers, we establish that: and: Thus, ). Note that Z h n and T h n are independent. It follows that the result of the theorem holds.
The convergence rate ofΦ(u) depends on the choice of h n , m n , and M n . The following theorem establishes the convergence rate of the integrated squared error ofΦ(u).

Conclusions
In this paper, the threshold and Fourier transform (inversion) techniques were employed to construct a new estimator of ruin probability for the spectrally negative Lévy process. The convergence rate of the integrated squared error (ISE) of the estimator was obtained when the jump component was the tempered α-stable subordinator. This shows that the ISE of the estimated ruin probability function is well controlled. Further work includes, but is not limited to deriving the asymptotic distribution of the proposed estimator and making statistical inference for ruin probability, under the framework that the risk model is a spectrally negative Lévy process with dividend strategy and investment. Furthermore, statistical inference for the Gerber-Shiu function and the dividend function are worthy of study.