Stochastic Process Theory and Its ApplicationsView this Special Issue
Threshold Estimation for a Spectrally Negative Lévy Process
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.
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 . 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 , Veraverbeke , Dermitzakis and Politis , and Li et al. . Others have been contributed to semiparametric and nonparametric estimation of the ruin probability. See, for example, Croux and Veraverbeke , Frees , Mnatsakanov et al. , Pitts , Politis , You et al. , and Zhang et al. .
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 from discrete time observations with step . The asymptotic framework is that tends to infinity and tends to zero while 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  have proposed a nonparametric estimator of ruin probability for pure jump Lévy process by Fourier (inversion) transform. The method of Zhang and Yang  has also been used by Shimizu and Zhang  to study the Gerber–Shiu function for Lévy subordinator. You and Cai  and Cai et al.  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 , Mancini , Shimizu , and Shimizu  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 Section 2. 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.
2.1. Risk Model
Letbe a spectrally negative Lévy process, where , , and is a standard Brownian motion; is a subordinator; Suppose that and are independent of each other.
The Laplace exponent of is denoted bywhere is a Lévy measure and satisfies .
Let be the initial surplus of an insurance company. The surplus at time is given bywhere is the rate of premium; represents the diffusion coefficient; and and denote the cumulative claims amount and a diffusion process.
2.2. Survival Probability
The infinite-time horizon survival probability is defined as follows:
In , Huzak et al. have given the following Pollaczek–Khinchin type formula for the survival probability,where , and is determined by the Laplace transform
By (5), the Laplace transform of is given by
3. Estimation of Survival Probability
In our work, we assume that and are unknown. In order to estimate , we need to estimate the Laplace transform of . Now, let us rewrite (7) as follows:where and .
Suppose that a discrete sample can be observed. Let and . Our interest is to estimate by a sample .
3.1. A Threshold Estimator of
In Shimizu , the author assumed that is a compound Poisson process. When , it is easy to judge a jump occurred if . As a result, can be an approximation of the jump size when and .
When is a compound Poisson process, we can give the following estimator of :
If has possibly a infinite number of jumps in each finite time interval, we still choose as an estimator of .
3.2. A Regularized Laplace Inversion Technique
In , Chauveau et al. have given the following regularized Laplace inversion technique.
Definition 1. Let be a constant. The regularized Laplace inversion is given byfor a function and , whereand .
By Definition 1, the regularized Laplace inversion technique is available for any functions. According to the proof of Proposition 3.3 in You and Cai , we know that . In order to apply Definition 1, we must amend .
Let for arbitrary fixed . Obviously, we haveTherefore, we will give an estimator of as follows:By Definition 1, we can construct an estimator of as follows:for suitable .
4. Asymptotic Property of Estimators
In this section, we will study the asymptotic property for those estimators which are proposed in Section 3. In , You and Cai have given the asymptotic normality and consistency of . The following work will consider the asymptotic property of , , and .
Let , where is a Poisson process with constant intensity . The random variables ,… are and independent of . Let , , and .
Theorem 1. Suppose that the net profit condition hold. If , , , , and for some , then for , we have
Theorem 2. Suppose that has the first derivative such that is of the polynomial growth and the conditions of Theorem 1 are satisfied. Then, for , and any constant , we have
Now, we consider that the process contains lots of small jumps, i.e., an infinite number of jumps in each finite time interval. Theorem 3 will give the consistency for and .
Theorem 3. Suppose that the net profit condition hold. If , , , and for some as , then
Theorem 4. Suppose that has the first derivative such that is of the polynomial growth and the conditions of Theorem 3 are satisfied. Then, for , , and , we havewhere as .
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 by , then is a compound Poisson process where the Poisson intensity is and the individual claim sizes are exponentially distributed with mean .
The survival probability is given bywhere are negative roots of the following equation
We take , , , , and . Then, and .
First of all, we consider . In Figure 1, we plot the mean points with sample sizes , which are computed based on 5000 simulation experiments.
In Table 1, we give the data of and its errors with sample sizes .
Now, we consider the estimator of . In Figure 2, we plot the mean points with sample sizes , which are computed based on 5000 simulation experiments.
Next, we consider the estimator of with . In Figure 3, we plot the mean points with sample sizes , which are computed based on 5000 simulation experiments.
By Figure 3, the estimator is very close to the actual data of as . Thus, we will use the same method as of Cai et al.  and You and Cai  to simulate . In order to improve computational efficiency, we definewhere , and , and .
Now, we will show the performance of .
In Figure 4, we plot the mean points with sample sizes and , which are computed based on 5000 simulation experiments.
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.  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. , 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.  and Yu et al. . The Gerber–Shiu function and the aggregate dividends up to ruin will be estimated by some statistical methods.
The proof of Theorem 1.
Proof. First of all, we haveLet us now deal with the first term of (A.1),By Proposition 3.2 and Corollary 3.3 in Mancini ,Because of , as .
Thus, the first term tends to zero in probability.
The second term of (A.1) tends to zero in probability, sinceTherefore, we only need to compute the limit in distribution of the third term of (A.1).
By Mancini , we know , thenBecause the random variables , , and are independent to each other and are , we have (A.5) that tends to in distribution as . Thus, we can obtain (19). By (19) and Slutsky’s Theorem, it is easy to obtain (18).
Next, by (7) and (16), we haveBy Slutsky’s Theorem, (11), (9), and (A.6), we haveBy ,By the independent property of , , , and ,In order to prove Theorem 2, we need the following Lemma 1 (see Theorem 3.2 in ).
Lemma 1. Suppose that for a function with the derivative , , then
The proof of Theorem 2.
Proof. Using (17),Let . Because satisfies the polynomial growth,By Lemma 1,By (7), (12), (15), and (16), we haveUsing the proof of Propositions 2.3 and 2.4 in You and Cai , the right-hand side in (A.14) is bounded bywhereBy Theorem 1, it follows thatas .
Combining (A.15) and (A.17)–(A.19) yieldsBy in Chauveau et al. , (A.13), and (A.20), we haveWith an optimal , balancing the right-hand two terms in (A.21), the order becomes .
The proof of Theorem 3.
Proof. Since and (11),Let us consider the second term of (A.22),By the Paul Lévy law for the modulus of continuity of Brownian motion’s path (Vondraček and Shreve , p.114, Theorem 9.25) implies thatWe havesinceas and .
Moreover,since for , .
Now, let us consider the last term of the right-hand side of (A.22):In fact, ifthenso thatsince a.s., for small , , for all , thenas , since .
Now, let us consider the term:Because is a spectrally negative Lévy process, for any , we havesince is arbitrary, we can conclude that , as , .
By the law of large number, we know that the first term and thirst term of (A.22) are convergence to zero in probability.
By (7), (16), (22), Theorem 1, and continuous mapping theorem, it is easy to obtain (23).
Therefore, we obtain the result of Theorem 3.
The proof of Theorem 4.
Proof. By (17), we haveBy the proof of Theorem 2, we may conclude thatwhereBy Theorem 3 and continuous mapping theorem, it follows thatCombining in Chauveau et al. , (A.36), and (A.39), we haveas .
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The authors were supported by the Postdoctoral Science Foundation of China (Grant no. 2018M642634) and the National Natural Science Foundation of China (Grant no. 11571198).
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