# Threshold Estimation for a Spectrally Negative Lévy Process

Authors: Honglong You; Chuncun Yin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Consider a spectrally negative Lévy process with unknown diffusion coefficient and Lévy measure and suppose that the high frequency trading data is given. We use the techniques of threshold estimation and regularized Laplace inversion to obtain the estimator of survival probability for a spectrally negative Lévy process. The asymptotic properties are given for the proposed estimator. Simulation studies are also given to show the finite sample performance of our estimator.


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## Body

## 1. Introduction

In actuarial science, it is an important topic to consider the ruin probability for some risk models. There are some methods for this topic, for example, the integro-differential equation technique, renewal theory, Laplace transform, martingale theory, and so on. For details, see the monograph of Asmussen and Albrecher [1]. These methods heavily depend on the knowledge of the risk model, which are usually unknown in practice. It is also known that explicit formula for the ruin probability is usually not available when we have no precise information on the risk model. In order to overcome this difficulty, many researchers have done a large amount of work and obtained lots of nice results. Some authors have considered the approximations, upper and lower bounds of the ruin probability. See, for example, Dufresne and Gerber [2], Veraverbeke [3], Dermitzakis and Politis [4], and Li et al. [5]. Others have been contributed to semiparametric and nonparametric estimation of the ruin probability. See, for example, Croux and Veraverbeke [6], Frees [7], Mnatsakanov et al. [8], Pitts [9], Politis [10], You et al. [11], and Zhang et al. [12].In practical situations, to get the data is much easier than to obtain the precise information on the risk model. In financial market, high frequency trading exists and a lot of high frequency trading data can be used to make statistical inference of the law of financial market. Using the data, we can estimate the survival probability by some statistical methods. In our work, we assume that the high frequency data is fromn discrete time observations with step hn. The asymptotic framework is that n tends to infinity and hn tends to zero while nhn tends to infinity. See, Comte and Genon-Catalot [13, 14]. For an insurance company, if the surplus has lots of small fluctuations, we assume that the surplus may be described by Lévy process. Our work will consider the survival probability for Lévy process. There are also some nice results for the ruin probability in Lévy process. For example, Zhang and Yang [15] have proposed a nonparametric estimator of ruin probability for pure jump Lévy process by Fourier (inversion) transform. The method of Zhang and Yang [15] has also been used by Shimizu and Zhang [16] to study the Gerber–Shiu function for Lévy subordinator. You and Cai [17] and Cai et al. [18] have constructed an estimator for the survival probability in a spectrally negative Lévy risk model by the regularized Laplace inversion technique. In the paper, we will use a threshold technique to study the survival probability. Mancini [19], Mancini [20], Shimizu [21], and Shimizu [22] have proposed the threshold technique for identifying the times when jumps larger than a suitably defined threshold occurred. Given a discrete record of observations, the technique may separate the contributions of the diffusion part and jump part of the risk model. Therefore, we can obtain more accurate data for the jump part of the risk model by the threshold technique. In this paper, we will give an estimator of the survival probability by the threshold technique and the regularized Laplace inversion technique. Our method will calculate the survival probability more accurately for a spectrally negative Lévy risk model.Here is a brief outline of this paper. We will introduce our risk model and define the survival probability in Section2. In Section 3, we will construct an estimator of survival probability for our risk model. Section 4 gives the asymptotic properties of the estimators. In Section 5, we will do some simulations to show the finite sample size performance of the estimators. Finally, some conclusions are given in Section 6. All the technical proofs are presented in Appendix.


## 2. Preliminaries

### 2.1. Risk Model

Let(1)Yt=ct+σWt−Jt,t≥0be a spectrally negative Lévy process, where c>0, σ>0, and W=Wt,t≥0 is a standard Brownian motion; J=Jt,t≥0 is a subordinator; Suppose that W and J are independent of each other.The Laplace exponent ofY=Yt,t≥0 is denoted by(2)ψYs=1tlnEesYt=cs+12σ2s2−∫0∞1−e−sxνdx,s>0,where ν is a Lévy measure and satisfies ∫0∞1∧xdx<∞.Letu>0 be the initial surplus of an insurance company. The surplus at time t is given by(3)Ut=u+Yt=u+ct+σWt−Jt,t≥0,where c is the rate of premium; σ represents the diffusion coefficient; and J and W denote the cumulative claims amount and a diffusion process.


### 2.2. Survival Probability

The infinite-time horizon survival probabilityΦu is defined as follows:(4)Φu=1−Pinf0≤t<∞Ut≤0U0=u.In [23], Huzak et al. have given the following Pollaczek–Khinchin type formula for the survival probability,(5)Φu=1−μ1c∑i=0∞ρiGi+1∗∗Hi∗u,u>0,where μ1=∫0∞xνdx, Hx=1/μ1∫0xνy,∞dy and G is determined by the Laplace transform(6)∫0∞e−sxdGx=cc+σ2/2s.By (5), the Laplace transform of Φu is given by(7)ℒΦs=∫0∞e−suΦudu=c−μ1cs+σ2/2s2−∫0∞1−e−sxνdx,s>0.


## 2.1. Risk Model

Let(1)Yt=ct+σWt−Jt,t≥0be a spectrally negative Lévy process, where c>0, σ>0, and W=Wt,t≥0 is a standard Brownian motion; J=Jt,t≥0 is a subordinator; Suppose that W and J are independent of each other.The Laplace exponent ofY=Yt,t≥0 is denoted by(2)ψYs=1tlnEesYt=cs+12σ2s2−∫0∞1−e−sxνdx,s>0,where ν is a Lévy measure and satisfies ∫0∞1∧xdx<∞.Letu>0 be the initial surplus of an insurance company. The surplus at time t is given by(3)Ut=u+Yt=u+ct+σWt−Jt,t≥0,where c is the rate of premium; σ represents the diffusion coefficient; and J and W denote the cumulative claims amount and a diffusion process.


## 2.2. Survival Probability

The infinite-time horizon survival probabilityΦu is defined as follows:(4)Φu=1−Pinf0≤t<∞Ut≤0U0=u.In [23], Huzak et al. have given the following Pollaczek–Khinchin type formula for the survival probability,(5)Φu=1−μ1c∑i=0∞ρiGi+1∗∗Hi∗u,u>0,where μ1=∫0∞xνdx, Hx=1/μ1∫0xνy,∞dy and G is determined by the Laplace transform(6)∫0∞e−sxdGx=cc+σ2/2s.By (5), the Laplace transform of Φu is given by(7)ℒΦs=∫0∞e−suΦudu=c−μ1cs+σ2/2s2−∫0∞1−e−sxνdx,s>0.


## 3. Estimation of Survival Probability

In our work, we assume thatσ and ν are unknown. In order to estimate Φu, we need to estimate the Laplace transform of Φu. Now, let us rewrite (7) as follows:(8)ℒΦs=1−ρ1/cψYs,s>0,where ρ=μ1/c and ψYs=cs+1/2σ2s2−∫0∞1−e−sxνdx.Suppose that a discrete sampleYn=Ytintin=ihn;i=0,1,2,…,n can be observed. Let Zi=Ytin−Yti−1n and hn=tin−ti−1n>0. Our interest is to estimate Φu by a sample Z1,Z2,…,Zn.


### 3.1. A Threshold Estimator ofℒΦs

In this part, we will construct an estimator ofℒΦs. If we can estimate ρ and ψYs in (8), the estimator of ℒΦs will be given by a plug-in device. By You and Cai [17], an estimator of ψYs has been given by(9)ψ^Ys=1hn1n∑k=1nesZk−1,s>0.Now, the following work is to estimateρ=μ1/c. By Shimizu [21, 22], we introduce the filter which is defined by(10)Dkn≔ω∈Ω;chn−Zk>hnb,Ckn≔ω∈Ω;chn−Zk≤hnb,where b is a positive constant.In Shimizu [21], the author assumed that J=Jt,t≥0 is a compound Poisson process. When b∈0,1/2, it is easy to judge a jump occurred if chn−Zk>hnb. As a result, chn−Zk can be an approximation of the jump size when chn−Zk>hnb and hn⟶0.WhenJ=Jt,t≥0 is a compound Poisson process, we can give the following estimator of ρ:(11)ρ^=1cnhn∑k=1nchn−ZkIDkn.IfJ=Jt,t≥0 has possibly a infinite number of jumps in each finite time interval, we still choose ρ^ as an estimator of ρ.Finally, by (8), (9), and (11), an estimator of ℒΦs is given by(12)ℒ^Φs=1−ρ^1/cψ^Ys,s>0.


### 3.2. A Regularized Laplace Inversion Technique

In [24], Chauveau et al. have given the following regularized Laplace inversion technique.Definition 1.
Letm>0 be a constant. The regularized Laplace inversion Lm−1:L20,∞⟶L20,∞. is given by(13)Lm−1gt=1π2∫0∞∫0∞Ψmyy−1/2e−tvygvdvdy,for a function g∈L20,∞ and t∈0,∞, where(14)Ψmy=∫0amcoshπxcosxlogydx,and am=π−1cosh−1πm>0.
By Definition1, the regularized Laplace inversion technique is available for any L20,∞ functions. According to the proof of Proposition 3.3 in You and Cai [17], we know that ℒ^Φs∉L20,∞. In order to apply Definition 1, we must amend ℒ^Φs.
LetΦθu:=e−θuΦu for arbitrary fixed θ>0. Obviously, we have(15)ℒΦθs=ℒΦs+θ,s>0.
Therefore, we will give an estimator ofℒΦθs as follows:(16)ℒ^Φθs≔ℒ^Φs+θ=1−ρ^1/cψY^s+θ,s>0.
By Definition1, we can construct an estimator of Φu as follows:(17)Φ^mu=eθuℒm−1ℒ^Φθs,u>0,θ>0,for suitable m>0.


## 3.1. A Threshold Estimator ofℒΦs

In this part, we will construct an estimator ofℒΦs. If we can estimate ρ and ψYs in (8), the estimator of ℒΦs will be given by a plug-in device. By You and Cai [17], an estimator of ψYs has been given by(9)ψ^Ys=1hn1n∑k=1nesZk−1,s>0.Now, the following work is to estimateρ=μ1/c. By Shimizu [21, 22], we introduce the filter which is defined by(10)Dkn≔ω∈Ω;chn−Zk>hnb,Ckn≔ω∈Ω;chn−Zk≤hnb,where b is a positive constant.In Shimizu [21], the author assumed that J=Jt,t≥0 is a compound Poisson process. When b∈0,1/2, it is easy to judge a jump occurred if chn−Zk>hnb. As a result, chn−Zk can be an approximation of the jump size when chn−Zk>hnb and hn⟶0.WhenJ=Jt,t≥0 is a compound Poisson process, we can give the following estimator of ρ:(11)ρ^=1cnhn∑k=1nchn−ZkIDkn.IfJ=Jt,t≥0 has possibly a infinite number of jumps in each finite time interval, we still choose ρ^ as an estimator of ρ.Finally, by (8), (9), and (11), an estimator of ℒΦs is given by(12)ℒ^Φs=1−ρ^1/cψ^Ys,s>0.


## 3.2. A Regularized Laplace Inversion Technique

In [24], Chauveau et al. have given the following regularized Laplace inversion technique.Definition 1.
Letm>0 be a constant. The regularized Laplace inversion Lm−1:L20,∞⟶L20,∞. is given by(13)Lm−1gt=1π2∫0∞∫0∞Ψmyy−1/2e−tvygvdvdy,for a function g∈L20,∞ and t∈0,∞, where(14)Ψmy=∫0amcoshπxcosxlogydx,and am=π−1cosh−1πm>0.
By Definition1, the regularized Laplace inversion technique is available for any L20,∞ functions. According to the proof of Proposition 3.3 in You and Cai [17], we know that ℒ^Φs∉L20,∞. In order to apply Definition 1, we must amend ℒ^Φs.
LetΦθu:=e−θuΦu for arbitrary fixed θ>0. Obviously, we have(15)ℒΦθs=ℒΦs+θ,s>0.
Therefore, we will give an estimator ofℒΦθs as follows:(16)ℒ^Φθs≔ℒ^Φs+θ=1−ρ^1/cψY^s+θ,s>0.
By Definition1, we can construct an estimator of Φu as follows:(17)Φ^mu=eθuℒm−1ℒ^Φθs,u>0,θ>0,for suitable m>0.


## 4. Asymptotic Property of Estimators

In this section, we will study the asymptotic property for those estimators which are proposed in Section3. In [17], You and Cai have given the asymptotic normality and consistency of ψ^Ys. The following work will consider the asymptotic property of ρ^, ℒ^Φθs, and Φ^m.IfJ=Jt,t≥0 is a compound Poisson process, we will give the following Theorems 1 and 2.LetJt=∑j=1Ntγj, where N=Nt,t≥0 is a Poisson process with constant intensity λ. The random variables γ1,γ2,γ3,… are i.i.d. and independent of N. Let μγ=Eγ1<∞, σγ2=Eγ12−μγ2<∞, and ℒγs=Ee−sγ1.Theorem 1.
Suppose that the net profit conditionc>λμγ hold. If b∈0,1/2, θ>0, limn⟶∞hn=0, limn⟶∞nhn=∞, and limn⟶∞nhn1+β=0 for some β∈0,1, then for n⟶∞, we have(18)ρ^−ρ⟶P0,(19)nhnρ^−ρ⟶DN0,1c2λμγ2+σγ2,(20)nhnℒ^Φθs−ℒΦθs⟶DN0,λμγ2+σγ2+σ2ψY2s+θ+μγ2ψY2s+2θψY4s+θ.Theorem 2.
Suppose thatΦu has the first derivative gu such that gu is of the polynomial growth and the conditions of Theorem 1 are satisfied. Then, for m=nhn/lognhn, u>0 and any constant B>0, we have(21)Φ^m−ΦB2=OP1lognhn,n⟶∞.Now, we consider that the processJ contains lots of small jumps, i.e., an infinite number of jumps in each finite time interval. Theorem 3 will give the consistency for ρ^ and ℒ^Φθs.Theorem 3.
Suppose that the net profit conditionc>μ1 hold. If b∈0,1/4, hn⟶0, nhn⟶∞, and nhn1+β⟶0 for some β∈0,1 as n⟶∞, then(22)ρ^−ρ⟶P0,(23)ℒ^Φθs−ℒΦθs⟶P0.Theorem 4.
Suppose thatΦu has the first derivative gu such that gu is of the polynomial growth and the conditions of Theorem 3 are satisfied. Then, for u>0, m>0, and B>0, we have(24)Φ^m−ΦB2≤m2oP1+am,n⟶∞,where am=O1/logm as m⟶∞.


## 5. Simulation

In this section, we will give some simulation studies to show the performance of our estimator with finite samples. The work is based on MATLAB. We do not pretend to find an optimal threshold function for each considered model.We assume that the Lévy measure is given byνdx=λ1/ηe−1/ηxdx, then J is a compound Poisson process where the Poisson intensity is λ and the individual claim sizes are exponentially distributed with mean η.The survival probability is given by(25)Φu=1−r1+1/η+2λη/σ2r1−r2er1u−r2+1/η+2λη/σ2r2−r1er2u,where r2<r1<0 are negative roots of the following equation(26)12σ2s+c−λs+1/η=0.We takec=λ=10, η=1/2, σ=5, b=1/4, and hn=n−4/5. Then, μ1=λη=5 and ρ=μ1/c=0.5.First of all, we considerρ^. In Figure 1, we plot the mean points with sample sizes n=1000,5000,8000,10000,and30000, which are computed based on 5000 simulation experiments.Figure 1
ρ^ with sample sizes n = 1000, 5000, 8000, 10000, and 30000.In Table1, we give the data of ρ^ and its errors with sample sizes n=1000,5000,8000,10000,and30000.Table 1
The data ofρ^ and its errors.
Sample sizesThe data ofρ^The actual data ofρErrorsn = 10000.48780.50.0122n = 50000.48760.50.0124n = 80000.50330.50.0033n = 100000.50190.50.0019n = 300000.49890.50.0011Now, we consider the estimator ofψY. In Figure 2, we plot the mean points with sample sizes n=500,1000,5000,8000,10000,30000,and50000, which are computed based on 5000 simulation experiments.Figure 2
The estimator ofψY with sample sizes n = 500, 1000, 5000, 8000, 10000, 30000, and 50000.Next, we consider the estimator ofℒΦθ with θ=0.075. In Figure 3, we plot the mean points with sample sizes n=5000,10000,30000,50000,and80000, which are computed based on 5000 simulation experiments.Figure 3
The estimator ofℒΦθ with sample sizes n = 5000, 10000, 30000, 50000, and 80000.By Figure3, the estimator ℒ^Φθ is very close to the actual data of ℒΦθ as n≥30000. Thus, we will use the same method as of Cai et al. [18] and You and Cai [17] to simulate Φu. In order to improve computational efficiency, we define(27)Φpu=euθ/π2∫0∞∫0∞e−usyℒ^Φθsn0Ψpyy−1/2dsdy,where ℒ^Φθsn0=1−1/cn0hn0∑k=1n0chn0−ZkIDkn0/1/chn01/n0∑k=1n0esZk−1, Ψpy=∫0apcoshπxcosxlogydx and ap=π−1cosh−1πp>0, and n0=30000.Now, we will show the performance ofΦpu.In Figure4, we plot the mean points with sample sizes n=30000 and p=100,500,800,1000,and3000, which are computed based on 5000 simulation experiments.Figure 4
True curve and mean curves with sample sizen = 30000 and p=100,500,800,1000,and3000.


## 6. Conclusion

In this paper, we use the threshold estimation technique and regularized Laplace inversion technique to construct an estimator of survival probability for a spectrally negative Lévy process. The rate of convergence for the estimator is a logarithmic rate. We adopt a method which is proposed in Cai et al. [18] to improve the speed in simulated calculation. The further work is to improve the speed of convergence for the estimator. We will combine the threshold estimation technique with the Fourier transform (inversion) technique, Fourier cosine series expansion method, and Laguerre series expansion method to construct an estimator of survival probability. These methods can be referred to Zhang and Hu [12, 25], Zhang and Su [26, 27], Yu et al. [28], and so on. We hope some further studies will be performed for the risk model with the barrier or threshold dividend strategy. See, e.g., Peng et al. [29] and Yu et al. [30]. The Gerber–Shiu function and the aggregate dividends up to ruin will be estimated by some statistical methods.



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