Papers by Yasutaka Shimizu
arXiv (Cornell University), Sep 13, 2022
We disucss a statistical estimation problem of an optimal dividend barrier when the surplus proce... more We disucss a statistical estimation problem of an optimal dividend barrier when the surplus process follows a Lévy insurance risk process. The optimal dividend barrier is defined as the level of the barrier that maximizes the expectation of the present value of all dividend payments until ruin. In this paper, an estimatior of the expected present value of all dividend payments is defined based on "quasi-process" in which sample paths are generated by shuffling increments of a sample path of the Lévy insurance risk process. The consistency of the optimal dividend barrier estimator is shown. Moreover, our approach is examined numerically in the case of the compound Poisson risk model perturbed by diffusion.
arXiv (Cornell University), Jun 19, 2022
We consider a surplus process of drifted fractional Brownian motion with the Hurst index > 1/2, w... more We consider a surplus process of drifted fractional Brownian motion with the Hurst index > 1/2, which appears as a functional limit of drifted compound Poisson risk models with correlated claims. This is a kind of representation of a surplus with a long memory. Our interest is to construct confidence intervals of the ruin probability of the surplus when the volatility parameter is unknown. We will obtain the derivative of the ruin probability w.r.t. the volatility parameter via Malliavin calculus, and apply the delta method to identify the asymptotic distribution of an estimated ruin probability.
arXiv (Cornell University), Jul 20, 2022
We consider parameter estimation of stochastic differential equations driven by a Wiener process ... more We consider parameter estimation of stochastic differential equations driven by a Wiener process and a compound Poisson process as small noises. The goal is to give a threshold-type quasi-likelihood estimator and show its consistency and asymptotic normality under new asymptotics. One of the novelties of the paper is that we give a new localization argument, which enables us to avoid truncation in the contrast function that has been used in earlier works and to deal with a wider class of jumps in threshold estimation than ever before.
SpringerBriefs in Statistics
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
ASTIN Bulletin
The survival energy model (SEM) is a recently introduced novel approach to mortality prediction, ... more The survival energy model (SEM) is a recently introduced novel approach to mortality prediction, which offers a cohort-wise distribution function of the time of death as the first hitting time of a “survival energy” diffusion process to zero. In this study, we propose a novel SEM that can serve as a suitable candidate in the family of prediction models. We also proposed a method to improve the prediction in an earlier work. We further examine the practical advantages of SEM over existing mortality models.
Statistical Inference for Stochastic Processes
We consider parameter estimation of stochastic differential equations driven by a Wiener process ... more We consider parameter estimation of stochastic differential equations driven by a Wiener process and a compound Poisson process as small noises. The goal is to give a threshold-type quasi-likelihood estimator and show its consistency and asymptotic normality under new asymptotics. One of the novelties of the paper is that we give a new localization argument, which enables us to avoid truncation in the contrast function that has been used in earlier works and to deal with a wider class of jumps in threshold estimation than ever before.
Statistics & Probability Letters
Japanese Journal of Statistics and Data Science
We consider stochastic differential equations (SDEs) driven by small Lévy noise with some unknown... more We consider stochastic differential equations (SDEs) driven by small Lévy noise with some unknown parameters, and propose a new type of least squares estimators based on discrete samples from the SDEs. To approximate the increments of a process from the SDEs, we shall use not the usual Euler method, but the Adams method, that is, a well-known numerical approximation of the solution to the ordinary differential equation appearing in the limit of the SDE. We show the consistency of the proposed estimators as well as the asymptotic distribution in a suitable observation scheme. We also show that our estimators can be better than the usual LSE based on the Euler method in the finite sample performance.
We study the problem of parametric estimation for continuously observed stochastic processes driv... more We study the problem of parametric estimation for continuously observed stochastic processes driven by additive small fractional Brownian motion with Hurst index 0
We study the problem of parameter estimation for discretely observed stochastic differential equa... more We study the problem of parameter estimation for discretely observed stochastic differential equations driven by small fractional noise. Under some conditions, we obtain strong consistency and rate of convergence of the least square estimator(LSE) when small dispersion coefficient converges to 0 and sample size converges to infty.
We consider M -estimation problems, where the target value is determined using a minimizer of an ... more We consider M -estimation problems, where the target value is determined using a minimizer of an expected functional of a Lévy process. With discrete observations from the Lévy process, we can produce a “quasi-path” by shuffling increments of the Lévy process, we call it a quasi-process. Under a suitable sampling scheme, a quasi-process can converge weakly to the true process according to the properties of the stationary and independent increments. Using this resampling technique, we can estimate objective functionals similar to those estimated using the Monte Carlo simulations, and it is available as a contrast function. The M -estimator based on these quasi-processes can be consistent and asymptotically normal.
The aim of this paper is to construct the confidence interval of the ultimate ruin probability un... more The aim of this paper is to construct the confidence interval of the ultimate ruin probability under the insurance surplus driven by a Lévy process. Assuming a parametric family for the Lévy measures, we estimate the parameter from the surplus data, and estimate the ruin probability via the delta method. However the asymptotic variance includes the derivative of the ruin probability with respect to the parameter, which is not generally given explicitly, and the confidence interval is not straightforward even if the ruin probability is well estimated. This paper gives the Cramér-type approximation for the derivative, and gives an asymptotic confidence interval of ruin probability.
ASTIN Bulletin, 2020
We propose a new approach to mortality prediction under survival energy hypothesis (SEH). We assu... more We propose a new approach to mortality prediction under survival energy hypothesis (SEH). We assume that a human is born with initial energy, which changes stochastically in time and the human dies when the energy vanishes. Then, the time of death is represented by the first hitting time of the survival energy (SE) process to zero. This study assumes that SE follows a time-inhomogeneous diffusion process and defines the mortality function, which is the first hitting time distribution function of the SE process. Although SEH is a fictitious construct, we illustrate that this assumption has the potential to yield a good parametric family of cumulative probability of death, and the parametric family yields surprisingly good predictions for future mortality rates.
Monte Carlo simulation is useful to compute or estimate expected functionals of random elements i... more Monte Carlo simulation is useful to compute or estimate expected functionals of random elements if those random samples are able to be generated from the true distribution. However, when the distribution has some unknown parameters, the samples must be generated from an estimated distribution with the parameters replaced by some estimators, which causes a statistical error in Monte Carlo estimation. This paper considers such a statistical error and investigates the asymptotic distributions of Monte Carlo-based estimators when the random elements are not only real valued, but also functional valued random variables. We also investigate expected functionals for semimartingales in detail. The consideration indicates that the Monte Carlo estimation can worsen when a semimartingale has a jump part with unremovable unknown parameters.
arXiv: Statistics Theory, 2020
We consider an estimation problem of expected functionals of a general random element that values... more We consider an estimation problem of expected functionals of a general random element that values in a metric space. If the functional forms an explicit function of some unknown parameters, we can estimate it by plugging-in a suitable estimator in to the function, and we can find the asymptotic distribution. However, if the functional is implicit in the parameters, it causes a problem of specifying asymptotic distribution. This paper gives a general condition to specify the asymptotic distribution even if the functional is implicit in the parameters, and further investigates it in detail when the random elements are semimartingales.
Insurance: Mathematics and Economics, 2016
Stochastic Processes and their Applications, 2017
We study parameter estimation for discretely observed stochastic differential equations driven by... more We study parameter estimation for discretely observed stochastic differential equations driven by small Levy noises. There have been many applications of small noise asymptotics to mathematical finance and insurance. Using small noise models we can deal with both applications and statistical inference. We do not impose Lipschitz condition on the dispersion coefficient function and any moment condition on the driving Levy process, which greatly enhances the applicability of our results to many practical models. Under certain regularity conditions on the drift and dispersion functions, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter based on discrete observations. The asymptotic distribution of the LSE in our general framework is shown to be the convolution of a normal distribution and a distribution related to the jump part of the driving Levy process. We present some simulation study on a two-factor financial model driven by stable noises.
The Gerber-Shiu function provides a unified framework for the evaluation of a variety of risk qua... more The Gerber-Shiu function provides a unified framework for the evaluation of a variety of risk quantities. Ever since its establishment, it has attracted constantly increasing interests in actuarial science, whereas the conventional research has been focused on finding analytical or semi-analytical solutions, either of which is rarely available, except for limited classes of penalty functions on rather simple risk models. In contrast to its great generality, the Gerber-Shiu function does not seem sufficiently prevalent in practice, largely due to a variety of difficulties in numerical approximation and statistical inference. To enhance research activities on such implementation aspects, we provide a full review of existing formulations and underlying surplus processes, as well as an extensive survey of analytical, semi-analytical and asymptotic methods for the Gerber-Shiu function, which altogether shed fresh light on its numerical methods and statistical inference for further developments. On the basis of an exhaustive collection of 207 references, the present survey can serve as an insightful guidebook to model and method selection from practical perspectives as well.
Consider a process satisfying a stochastic differential equation with unknown drift parameter, an... more Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent, but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data, and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was choosen suitably, and the noise was a L\'evy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.
We are interested in statistical inference for the finite-time ruin probability of an insurance s... more We are interested in statistical inference for the finite-time ruin probability of an insurance surplus whose claim process has a long-range dependence. As an approximated model, we consider a surplus driven by a fractional Brownian motion with the Hurst parameter H > 1/2. We can compute the ruin probability via the Monte Carlo simulations if some unknown parameters in the model are decided. From discrete samples, we estimate those unknowns, by which an asymptotically normal estimator of the ruin probability is computed. An expression of the asymptotic variance is given via the Malliavin Calculus in the estimable form. As a result, we can construct a confidence interval of the finite-time ruin probability. Since the ruin is usually rare event, an importance sampling technique is sometimes usuful in computation in practice.
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Papers by Yasutaka Shimizu