Suppose that we are given discrete-time but high-frequency observation from a solution to the one-dimensional diffusion with jumps described by
where the ingredients are given as follows.
is a standard Wiener process and a compound Poisson process associated with the Lévy measure
for some probability distribution. Throughout we assume that .
The sampling times fulfills that
where the terminal sampling time ; hereafter, we will largely abbreviate “” from the notation like and .
A well-known approach to estimate
contains the jump component if for a fixed jump-detection threshold , and estimate after removing such increments. It is shown that for a good satisfying a suitable rate, the estimator of has asymptotic normality at the same rate as diffusion models. Hence the method asymptotically achieves both the estimation of and the jump detection in observed data, while finite-sample performance of the threshold method strongly depends on the value of . Unfortunately, a data-adaptive and quantitative choice of the threshold in the jump-detection filter is a subtle and sensitive problem, and still remains as an annoying problem in practice; see , , as well as the references therein. Such problem can also be seen in other jump detection methods such as .
The primary objective of this paper is to formulate an intuitively easy-to-understand strategy, which can simultaneously estimate and detect jumps without any precise calibration of a jump-detection threshold. For this purpose, we utilize the approximate self-normalized residuals  based on the Gaussian quasi maximum likelihood estimator (GQMLE), which makes a classical Jarque-Bera type test  adapted to our model. More specifically, the hypothesis test whose significance level is
is constructed by the following manner: let the null hypothesis be of “no jump component” :
against the alternative hypothesis of “non-trivial jump component”:
The rest of this paper is organized as follows: in Section 2, we will give a briefly summary of the GQMLE, the approximate self-normalized residuals and the Jarque-Bera test for our model. Section 3 provides the specific recipe of ours and an alternative estimator to GQMLE in order to reduce computational load. At last, we will show some numerical experiments of our method.
In this section, we briefly review the construction of GQMLE, self-normalized residual, and Jarque-Bera statistics with its theoretical behavior. Given any function on , we write
We denote by the image measure of associated with the parameter value , and by the Poisson random measure associated with .
Suppose that the null hypothesis
is true for a moment; namely the underlying model is a diffusion process. Then, for the estimation of, we can make use of the Gaussian quasi-(log-)likelihood
where denotes the standard normal density and
This quasi-likelihood is constructed based on the local-Gauss approximation of the transition probability by under , and lead to the Gaussian quasi-maximum likelihood estimator (GQMLE) defined by any element
It is well known that the asymptotic normality holds true  under suitable regularity conditions:
Here denotes the invariant measure of .
To see whether a working model fits data well or not, diagnosis based on residual analysis is often done. Based on the GQMLE,  formulated Jarque-Bera normality test based on self-normalized residuals for our model. Define the self-normalized residual statistic by:
where and . Making use of , we define Jarque-Bera type statistic by
Then, Jarque-bera normality test for our model is justified by the following sense:
(cf. [6, Theorem 3.1 and Theorem 4.1]) Under the suitable regularity conditions, we have the followings:
Under , we have ;
Under , we have .
3. Proposed strategy
For brevity we write
Let be a small number. We propose the iterative jump detection procedure based on the Jarque-Bera type test below.
Set , and let be empty set.
Calculate the modified GQMLE (MGQMLE, for short) by:
where . Define the following statistics:
Building on the MGQMLE and the above ingredients, (re-)construct the following modified self-normalized residuals and Jarque-Bera type statistics :
If , then pick out the interval number
add to , and return to Step 1; otherwise, set the number of jumps , and go to Step 3.
If , regard that there is no jump; otherwise, the detected jumps are (they are in descending order). Finally, set as the estimator of .
By using its intensity parameter , the number of jumps of a compound Poisson process is expressed as . Thus, as the terminal time gets larger and larger, the iteration number of our proposed methodology should also be large. In such case or the case where seemingly several jumps do exist, we could instead start from -th stage for some which conveniently enables us to “skip” first some redundant stages.
In practice, the size of “last-removed” increment would be used as the threshold for detecting jumps for future observations: with the value in hand, for future observations we regard that a jump occurred over if
Our method enables us to divide the set of the whole increments into the following two categories:
“One-jump” group , and
“No-jump” group .
Our method conducts the estimation of the drift and diffusion part of based on continuously joined up data computed from the no-jump group pairs:
Also, we may estimate the jump part by the members of one-jump group; namely we think that the sequence under being i.i.d. with common jump distribution of the compound Poisson process .
To reduce the computational load of the calculating the GQMLE, one can alternatively use the stepwise estimator defined by:
and its modified version can similarly be defined. Under the null hypothesis being true, the limit distribution of is shown to be equivalent to that of (cf. ). Moreover, computation of the GQMLE and MGQMLE may become much less time-consuming one when the coefficients are of certain tractable forms: let and be the dimension of and , respectively, and suppose that the diffusion coefficient and the drift function can be written by suitable functions and as
where denotes the -th element of for every vector
for every vector. Then the stepwise estimator is given by
where and . What is important from these expressions is that the modified version of can be calculated simply by removing the corresponding indices from the sum without repetitive numerical optimizations, thus reducing the computational time to a large extent.
4. Asymptotic property of the MGQMLE
In this section, we look at the asymptotic properties of the MGQMLE for the following toy model:
where is a compound Poisson process expressed as
In this expression, and denote a Poisson process whose intensity parameter is
and i.i.d random variables, respectively. Recall that the observationsare obtained in , . To deduce the asymptotic properties of the MGQMLE, we introduce some assumptions below.
, and there exists positive deterministic sequence satisfying the conditions
For any , we have
the number of jump removal .
The following theorem ensures a consistency property of the MGQMLE:
If Assumption 4.1 holds, then we have
for each and .
5. Numerical experiments
We consider the following SDE model:
where . Here we set the true values as and . Under the conditions where:
, and selected with equal probabilities.
Here, we set number of jumps fixed just for numerical comparison purpose. Then the performance of our method is given in the table 1.
|before jump removal||0.31||2.00|
|after jump removal||0.30||1.00|
The performance of the GQMLE and MGQMLE: the mean is given with the standard deviation in parenthesis.
, the estimation accuracy is drastically improved by our method. In this example, we set the jump distribution symmetric, thus the improvement of the estimation of the drift parameter is small compared with that of the diffusion parameter; the amount of improvement is expected to be much more significant when the jump distribution is skewed.
This work was supported by JST, CREST Grant Number JPMJCR14D7, Japan.
6. Appendix: proof of Theorem 4.2
Let Assumption 4.1 hold throughout this section. First, we prove two lemmas.
Let denote jump times of . Then we have
Since the increments of the jump times of Poisson process independently obey the exponential distribution with mean, it follows that
For convenience, we hereafter write
Lemma 6.2 implies that all increments containing jumps are correctly picked up as long as . Similarly, we can derive
Proof of Theorem 4.2
We introduce the following events:
Taking the lemmas into consideration, we can split as
Since , it suffices to show and . From now on, for an event we denote by the indicator function of :
First we focus on the estimate of . By virtue of the foregoing discussion, we have the following expression:
Hence it follows that
The law of large numbers for triangular sequences implies that
Again applying (6.1), we have
Let us now move on to the estimates of . From the representation
and the central limit theorem, it follows that
Hence if , we have