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.

ジャンプ型の確率微分方程式による資産モデルを用いた保険数理における破産確率、及びその一般化であるGerber-Shiu関数をによるリスク尺度の構築を行った。特に、経時的に変化するリスクを評価する動的リスク尺度の構築と、その数学的正当化を行った。次に、これらのリスク尺度に死亡リスク評価を含める目的で、死亡率予測モデルの予測精度改善に関する研究を行った。信頼性理論に基づいた小地域の死亡率推定に関する新しい方法論を提案したが、予測精度の向上については課題を残した。加えて、Gerber-Shiu関数に対する統計推測理論の研究も行った。これは上記で構築したリスク尺度の実務的応用のために必須のものである

有限時間高頻度観測される点過程の強度過程が伊藤過程であるときに，その相関構造推定量を構成し，漸近挙動を解明した．これはリード・ラグ推定やプライスモデリングの基礎となる．点過程回帰モデルに対して，統計的確率場と多項式型大偏差不等式を基礎とする擬似尤度解析を構築し，とくにロングタームの場合にそれを与えた．オーダーブックのダイナミックスを表現するモデルを提案し，実証分析を行った

この研究プロジェクトでは、マクロ経済リスク、ミクロ金融リスク、保険リスク、など現代経済に潜む様々な経済リスク・金融リスクの統計的分析において重要な課題について、特に確率過程(確率過程)・数理統計学的側面と統計的応用について検討した。主な研究内容としては、マクロ経済リスクについては確率過程計量モデルによる希な現象の統計分析、金融リスクについてはリスク尺度の理論と統計学、保険リスクについては人口・生命保険の応用及び損害保険の確率論と応用、などが挙げられる。特に高頻度金融データを用いたリスク評価のための新たな統計モデルの開発、マクロ経済リスク分析のための新たな点過程統計モデルを新たに開発した

有限時間高頻度観測における確率回帰モデルのボラティリティの疑似尤度解析において，確率場の非退化性を保証する解析的および幾何学的判定条件を与え，擬似最尤推定量および擬似ベイズ推定量の漸近混合正規性と積率収束を証明した．有限時間非同期高頻度観測下で疑似尤度解析を構成した．極限が混合正規となるマルチンゲールの漸近展開が確立した．これは伝統的な漸近展開理論の枠を越える新しい極限定理である．応用としてp-変動の漸近展開が導出された．マイクロストラクチャーノイズ除去を伴う分散推定量の漸近展開の研究が進展した．情報量規準sVICや，ソフトウエア開発のための基礎理論研究を行った．

有限時間高頻度観測における確率回帰モデルのボラティリティの疑似尤度解析において，確率場の非退化性を保証する解析的および幾何学的判定条件を与え，擬似最尤推定量および擬似ベイズ推定量の漸近混合正規性と積率収束を証明した．有限時間非同期高頻度観測下で疑似尤度解析を構成した．極限が混合正規となるマルチンゲールの漸近展開が確立した．これは伝統的な漸近展開理論の枠を越える新しい極限定理である．応用としてp-変動の漸近展開が導出された．マイクロストラクチャーノイズ除去を伴う分散推定量の漸近展開の研究が進展した．情報量規準sVICや，ソフトウエア開発のための基礎理論研究を行った

古典的保険破産解析の一般化として，レヴィ型リスクモデルに対する一般化Gerber-Shiu解析を展開した．主要結果としては，代表的な破産関連リスクであるGerber-Shiu関数を，サープラスの積分型汎関数として一般化し，その再生型積分方程式や，レヴィ過程のスケール関数による表現定理を導出した．また，インフレーション型リスクモデル(確率微分方程式モデル)に対する破産理論を展開し，破産確率評価や再保険戦略などの具体的問題を解いた．さらに統計的方法として，破産確率の漸近展開近似や，データに基づいた推測理論を構築し，推定量の誤差評価やその収束率を求めた．これらは数値実験によりその有効性が確認された

金融や保険の資産モデルなどに応用される確率モデルに含まれる未知パラメータに対する,データに基づく実際的な推定手法を開発し,その理論的な正当性を数学的に証明した.その手法の計算機への実装も進行中である

少人数の定例研究会の開催,中規模研究集会の計画と開催,そして国際研究集会の編成と開催を通じて,統計科学,情報科学,社会科学(特に科学哲学を含む)において情報交換を積極的に行った.具体的な研究成果として,共分散選択のロバスト推定法の開発,非正規性の利用による因果の方向の同定,2×2分割表と潜在変数モデルにおける無視不可能な欠測に対する新たな分析方法の開発,条件付き確率と科学的証拠の関係の明確化などが得られた

飛躍構造が複合ポアソン的なジャンプ拡散過程に対する,離散観測に基づく種々の統計推測理論が構築された.飛躍判別フィルターのデータに基づく構成法がそれら理論の鍵となるが,ある特殊なモデルに対してその構成に関わる超パラメータの選択法と理論的正当性が得られた.これらの成果は,近年の金融保険数理で用いられる応用モデルの実装上,欠かせない結果であり,次のステップとなる無限ジャンプ型のモデルに対す推測論の考察の上で有益な成果と言える

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．