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 chosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.

We study parameter estimation for discretely observed stochastic differential equations driven by small Lévy noises. We do not impose Lipschitz condition on the dispersion coefficient function σ and any moment condition on the driving Lévy 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 parameter when ε→0 and n→∞ simultaneously. We present some simulation study on a two-factor financial model driven by stable noises.

Consider an insurance surplus process driven by a Levy subordinator, which is observed at discrete time points. An estimator of the Gerber-Shiu function is proposed via the empirical Fourier transform of the Gerber-Shiu function. By evaluating its mean squared error, we show the L-2-consistency of the estimator under the assumption of high-frequency observation of the surplus process in a long term. Simulation studies are also presented to show the finite sample performance of the proposed estimator. (C) 2017 Elsevier B.V. All rights reserved.

In both the past literature and industrial practice, it was often implicitly used without any justification that the classical strong law of large numbers applies to the modeling of equity-linked insurance. However, as all policyholders' benefits are linked to common equity indices or funds, the classical assumption of independent claims is clearly inappropriate for equity-linked insurance. In other words, the strong law of large numbers fails to apply in the classical sense. In this paper, we investigate this fundamental question regarding the validity of strong laws of large numbers for equity-linked insurance. As a result, extensions of classical laws of large numbers and central limit theorem are presented, which are shown to apply to a great variety of equity-linked insurance products.

The Markov additive process (MAP) has become an increasingly popular modeling tool in the applied probability literature. In many applications, quantities of interest are represented as functionals of MAPs and potential measures, also known as resolvent measures, have played a key role in the representations of explicit solutions to these functionals. In this paper, closed-form solutions to potential measures for spectrally negative MAPs are found using a novel approach based on algebraic operations of matrix operators. This approach also provides a connection between results from fluctuation theoretic techniques and those from classical differential equation techniques. In the end, the paper presents a number of applications to ruin-related quantities as well as verification of known results concerning exit problems. (C) 2014 Elsevier B.V. All rights reserved.

The YUIMA Project is an open source and collaborative effort aimed at developing the R package yuima for simulation and inference of stochastic differential equations. In the yuima package stochastic differential equations can be of very abstract type, multidimensional, driven by Wiener process or fractional Brownian motion with general Hurst parameter, with or without jumps specified as Ĺevy noise. The yuima package is intended to offer the basic infrastructure on which complex models and inference procedures can be built on. This paper explains the design of the yuima package and provides some examples of applications.

This paper presents an asymptotic expansion of the ultimate ruin probability under Levy insurance risks as the loading factor tends to zero. The expansion formula is obtained via the Edgeworth type expansion for compound geometric distributions. We give higher-order expansion of the ruin probability, any order of which is available in explicit form, and discuss a certain type of validity of the expansion. We shall also give applications to evaluation of the VaR-type risk measure due to ruin, and the scale function of spectrally negative Levy processes.

In a variety of insurance risk models, ruin-related quantities in the class of expected discounted penalty function (EDPF) were known to satisfy defective renewal equations that lead to explicit solutions. Recent development in the ruin literature has shown that similar defective renewal equations exist for a more general class of quantities than that of EDPF. This paper further extends the analysis of this new class of functions in the context of a spectrally negative Lévy risk model. In particular, we present an operator-based approach as an alternative analytical tool in comparison with fluctuation theoretic methods used for similar quantities in the current literature. The paper also identifies a sufficient and necessary condition under which the classical results from defective renewal equation and those from fluctuation theory are interchangeable. As a by-product, we present a series representation of scale function as well as potential measure in terms of compound geometric distribution. © 2012 Springer Science+Business Media, LLC.

We study the first-passage time over a fixed threshold for a pure-jump subordinator with negative drift. We obtain a closed-form formula for its survival function in terms of marginal density functions of the subordinator. We then use this formula to calculate finite-time survival probabilities in a structural model for credit risk, and thus obtain a closed-form pricing formula for a single-name credit default swap (CDS). This pricing formula is well calibrated on market CDS quotes. In particular, it explains why the par CDS credit spread is not negligible when the maturity becomes short. (C) 2013 Elsevier B.V. All rights reserved.

We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Levy noises. We do not impose any moment condition on the driving Levy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient epsilon -> 0 and n -> infinity simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to the jump part of the Levy process. Moreover, we briefly remark that our methodology can be easily extended to the more general case of semi-martingale noises. (C) 2013 Elsevier Inc. All rights reserved.

A specific form of stochastic differential equation with unknown parameters are considered. We do not necessarily assume ergodicity or recurrency, and any moment conditions for the true process, but some tightness conditions for an information-like quantity. The interest is to estimate the parameters from discrete observations the step size of which tends to zero. Consistency and the rate of convergence of proposed estimators are presented. The rate is deduced naturally from the rate for the information-like quantities. © 2010 The Institute of Statistical Mathematics, Tokyo.

A non-parametric estimator of the Gerber-Shiu function is proposed for a risk process with a compound Poisson claim process plus a diffusion perturbation; the Wiener-Poisson risk model. The estimator is based on a regularized inversion of an empirical-type estimator of the Laplace transform of the Gerber-Shiu function. We show the weak consistency of the estimator in the sense of an integrated squared error with the rate of convergence. © 2012 Taylor and Francis Group, LLC.

Consider non-recurrent Ornstein-Uhlenbeck processes with unknown drift and diffusion parameters. Our purpose is to estimate the parameters jointly from discrete observations with a certain asymptotics. We show that the likelihood ratio of the discrete samples has the uniform LAMN property, and that some kind of approximated MLE is asymptotically optimal in a sense of asymptotic maximum concentration probability. The estimator is also asymptotically efficient in ergodic cases.

We consider a generalized risk process which consists of a subordinator plus a spectrally negative Lévy process. Our interest is to estimate the expected discounted penalty function (EDPF) from a set of data which is practical in the insurance framework. We construct an empirical type estimator of the Laplace transform of the EDPF and obtain it by a regularized Laplace inversion. The asymptotic behavior of the estimator under a high frequency assumption is investigated. © 2011 Allerton Press, Inc.

Threshold estimation is one of the useful techniques in the inference for jump-type stochastic processes from discrete observations. In this method, a jump-discriminant filter is used to infer the continuous part and the jump part separately. Although there are several choices for the filter, statistics constructed via filters are often sensitive to the choice. This paper presents some numerical procedures for selecting a suitable filter based on observations.

Given discrete samples from Ornstein-Uhlenbeck processes, we consider two kinds of approximated MLE's for the drift parameter, which are asymptotically efficient in ergodic case. Our interest is the rate of convergence of those estimators when the process is non-recurrent. We add a remark when the underlying process has a slightly more general drift. © 2009 Elsevier B.V. All rights reserved.

We introduce a new aspect of a risk process, which is a macro approximation of the flow of a risk reserve. We assume that the underlying process consists of a Brownian motion plus negative jumps, and that the process is observed at discrete time points. In our context, each jump size of the process does not necessarily correspond to the each claim size. Therefore our risk process is different from the traditional risk process. We cannot directly observe each jump size because of discrete observations. Our goal is to estimate the adjustment coefficient of our risk process from discrete observations. © 2008 Elsevier B.V. All rights reserved.

We deal with parametric inference and selection problems for jump components in discretely observed diffusion processes with jumps. We prepare several competing parametric models for the Levy measure that might be misspecified. and select the best model from the aspect of information criteria. We construct quasi-information criteria (QIC), which are approximations of the information criteria based on continuous observations. (C) 2008 Elsevier B.V. All rights reserved.

We consider a semimartingale with jumps that are driven by a finite activity Lévy process. Suppose that the Lévy measure is completely unknown, and that the jump component has a Markovian structure depending on unknown parameters. This paper concentrates on estimating the parameters from continuous observations under the nonparametric setting on the Lévy measure. The estimating function is proposed by way of nonparametric approach for some regression functions. In the end, we can specify jumps of the underlying Lévy process and estimate some Lévy characteristics jointly. © 2008 Allerton Press, Inc.

In the inference for jump-diffusion processes, we often need to get the information of the jump part and of the continuous part separately from the data. Although some asymptotic theories have been studied on this issue, a practical interest is the inference from finitely many discrete samples. In this paper we propose a numerical procedure to construct a filter to judge whether or not a jump occurred from finite samples. The paper includes a discussion about the validity of the procedure.

In this paper, we consider a multidimensional diffusion process with jumps whose jump term is driven by a compound Poisson process. Let a(x,θ) be a drift coefficient, b(x,σ) be a diffusion coefficient respectively, and the jump term is driven by a Poisson random measure p. We assume that its intensity measure q θ has a finite total mass. The aim of this paper is estimating the parameter α = (θ,σ) from some discrete data. We can observe n+1 data at t i n = ih n, 0 ≤ i ≤ n. We suppose h n → 0, nh n → ∞, nh n 2 → 0. © Springer 2006.

We study parametric inference for multidimensional stochastic differential equations with jumps from some discrete observations. We consider a case where the structure of jumps is mainly controlled by a random measure which is generated by a Lévy process with a Lévy measure f θ(z)dz, and we admit the case f θ(z)dz=∞ in which infinitely many small jumps occur even in any finite time intervals. We propose an estimating function under this complicated situation and show the consistency and the asymptotic normality. Although the estimators in this paper are not completely efficient, the method can be applied to comparatively wide class of stochastic differential equations, and it is easy to compute the estimating equations. Therefore, it may be useful in applications. We also present some simulation results for some simple models. © Springer 2006.

We constructed a risk measure based on the ruin probability and the Gerber-Shiu function, which is a generalization of the ruin probability, under the surplus model by stochastic differential equation with jumps. In particular, we considered “dynamic” risk measures in order to measure risks that varies in time with their mathematical justification. Next, we investigated a mortality prediction model for the purpose of including a mortality risk into our risk measure. We proposed a new methodology to improve the small area estimation of mortality by applying Credibility Theory. However the performance of the method is not satisfactory and we will leave it as a future work. Furthermore we investigated the statistical inference for the Gerber-Shiu function, which is necessary for application of risk measures in practice

An estimator for correlation between two intensity processes was proposed and limit theorems were proved when the intensities diverge under finite time horizon. These results form the basis for lead-lag estimation and price modeling. For a point process regression model, we constructed Quasi Likelihood Analysis based on statistical random fields and polynomial type large deviation inequalities, especially in long-term asymptotics. Models of limit order book dynamics were proposed and fitted to real data

This research project has studied important issues of economic risk analysis including financial risk and insurance risk in modern society and economy. We have studied various scientific and systematic aspects of economic risk analysis the perspective of probability as well as statistics. The project members have presented research results at various international as well as domestic academic meetings, and published some books and many academic papers.For financial risk and insurance risk analysis we have investigated the statistical risk theory and mathematical finance. For economic risk, we have investigated the point process approach, which would make a new risk analysis of economic rare events. We also made some progress on the consistent statistical analysis of micro analysis (or high frequency risk analysis) and macro analysis (or low frequency risk analysis)

The quasi likelihood analysis was constructed for a stochastic regression model of volatility based on high frequency data in the finite time horizon, and an analytic criterion and a geometric criterion for non-degeneracy of the statistical random field associated with the quasi likelihood function were provided. The asymptotic mixed normality and the convergence of moments were proved. A quasi likelihood analysis was developed for a non-synchronously observed stochastic differential equation. Asymptotic expansion for a martingale with mixed normal limit was established. It is a new limit theorem beyond the frame of the present theory of asymptotic expansion for ergodic processes. The martingale expansion was applied to the p-variation. Studies of the asymptotic expansion of volatility estimators under microstructure noise have been developed. The spot volatility information criterion sVIC was proposed, and the fundamentals for developing computer software were studied.

The quasi likelihood analysis was constructed for a stochastic regression model of volatility based on high frequency data in the finite time horizon, and an analytic criterion and a geometric criterion for non-degeneracy of the statistical random field associated with the quasi likelihood function were provided. The asymptotic mixed normality and the convergence of moments were proved. A quasi likelihood analysis was developed for a non-synchronously observed stochastic differential equation. Asymptotic expansion for a martingale with mixed normal limit was established. It is a new limit theorem beyond the frame of the present theory of asymptotic expansion for ergodic processes. The martingale expansion was applied to the p-variation. Studies of the asymptotic expansion of volatility estimators under microstructure noise have been developed. The spot volatility information criterion sVIC was proposed, and the fundamentals for developing computer software were studied

As a generalization of the classical insurance ruin theory, we investigated a generalized Gerber-Shiu analysis under Levy insurance risk models. Main results are an extension of the ruin-related risk (Gerber-Shiu function) to a integral type functional of the insurance surplus, the derivation of its renewal type equation, and a representation theorem by a scale function for a spectrally negative Levy process. Moreover, we studied an inflation risk model written by a stochastic differential equation, and gave a bound of ruin probability and an optimal strategy of a reinsurance. In statistical analysis, we gave an approximation by the Edgeworth type expansion of ruin probability, inference for the Gerber-Shiu function from a discrete samples, and investigated the statistical error with the rate of convergence. We also show by simulations that these methodologies numerically work well

We developed some statistical methodologies to estimate unknown parameters in stochastic models, which is used in financial and insurance applications. The estimators are based on a set of practical data, and has a mathematical validity. The implementation to a computational package is also ongoing

Active academic exchanges among researchers in statistical, informatics and social sciences, including philosophy of science in particular, have been made through hosting colloquiums of a few speakers regularly, symposiums of 10 to 15 talks several times a year, and international meetings. Some of our fruits obtained with the help of the governmental scientific research grant are i) a new robust methodology of the covariance selection, ii) a new method for determining direction of causation using nonnormality, ii) new estimation methods for data with nonignorable missing values in a 2 times 2 contingency table and a latent-variable model, and iii) clarification between conditional probability and scientific evidence

本研究では，損害保険数理に現れる保険ポートフォリオの破産問題の数学的一般化とその統計推測を主題として，(1) 資産モデルの一般化； (2) 一般化破産関連リスクの定式化と解析評価；(3)一般化破産リスクに対する統計推測理論の構築，に焦点を当てて研究を行った．これに対して，(1)では資産モデルを一般のレヴィ過程に拡張し，(2)ではGerber-Shiu関数を含むレヴィ過程の積分形汎関数を提案，その微分・積分方程式の導出に成功した．また(3)については，離散的な資産データによるノンパラメトリック推定量を提案し，そのL2誤差評価を与えた．