研究紹介
確率過程の極限的性質に興味があります．一次元ランダムウォークに対するレヴィの逆正弦法則について，一般の一次元拡散過程に対する拡張，および平均的中立不動点を持つランダム力学系に対する拡張，を研究してきました．また，処罰問題と呼ばれる，特定の挙動を抑止する条件付け長時間極限の問題に関連して，レヴィ過程に対する様々な形の極限定理を研究してきました．
(r05) 一次元拡散過程. 特集「伊藤清と確率論」, 数学セミナー2015年9月号, 20--25, 2015.
(r04) Something may appear from nothing... Kyoto University Research Activities Vol.3 No.2, pp. 22, 2013.
(r03) Excursion theoryとその応用(サーベイ講演), 確率論シンポジウム, 2011.
(r02) 確率的な動きの科学. 神戸大学研究最前線, vol. 14, pp. 16--17, 2010.
(r01) 一次元拡散過程のexcursionと極限定理. 日本数学会秋季総合分科会 統計数学分科会 特別講演, 2006.
過去のセミナーテキスト・学部
O. Kallenberg. Foundations of Modern Probability. Springer.
E. Çinlar. Probability and Stochastics. Springer.
D. Williams. Probability with Martingales. Cambridge University Press.
過去のセミナーテキスト・修士・博士
O. Kallenberg. Foundations of Modern Probability. Springer.
A. E. Kyprianou. Fluctuations of Levy Processes with Applications Introductory Lectures. Springer.
Ph. E. Protter. Stochastic Integration and Differential Equations. Springer.
J. Aaronson. An Introduction to Infinite Ergodic Theory. AMS.
A. Swishchuk and S. Islam. Random Dynamical Systems in Finance. CRC Press.
A. Boyarsky and P. Góra. Laws of Chaos Invariant Measures and Dynamical Systems in One Dimension. Birkhäuser.
V. S. Barbu and N. Limnios. Semi-Markov Chains and Hidden Semi-Markov Models toward Applications: Their Use in Reliability and DNA Analysis. Springer.
S. Asmussen and H. Albrecher. Ruin Probabilities. World Scientific.
指導学生の博士学位論文のタイトル
A unifying approach to non-minimal quasi-stationary distributions for one-dimensional diffusions (2022)
Functional limit theorem for occupation time processes of intermittent maps (2020)
Generalized scale functions and refracted processes (2019)
Convergence of stochastic processes on varying metric spaces (2016)
指導学生の修士論文のタイトル
Nonparametric estimations for discrete-time semi-Markov chains (2023)
マルコフ更新定理 (2023)
Local time penalizations with various clocks for Lévy processes (2022)
Fluctuation scaling limit of inverse local times of jumping-in diffusions (2019)
Arcsine law for a piecewise linear random map (2019)
破産確率の様々な側面について (2019)
損保数理に用いられる分布族の最尤推定とその一致性 (2019)
Limit theorems for occupation ratio measures of infinite ergodic transformations (2018)
切替拡散過程の分布安定性 (2018)
Generalized refracted Lévy process and its application to exit problem (2016)
切替拡散過程の再帰性 (2016)
マルコフ切替を持つ拡散過程の指数安定性 (2016)
マルコフ連鎖における熱流及び勾配流 (2013)
飛躍型確率過程に対する閾値推定法 (2013)
Convergence of Markov processes on ultrametric spaces and a projection Markov property (2013)
安定分布に従う確率変数の級数表示と漸近挙動 (2012)
Filtering theory for Markov chains in discrete time and space (2012)
Brussの最適停止定理について (2011)
Gauss過程に対するweak Itô formulaについて (2011)
分布の意味での逆関数のいくつかの例について (2010)
負の無限大からの時間発展を表す確率方程式 (2010)
ランダムウォークのノルム過程に関する0-1法則 (2009)
指導学生が関係した雑誌等掲載論文
Local time penalizations with various clocks for Lévy processes (2023)
A unifying approach to non-minimal quasi-stationary distributions for one-dimensional diffusions (2022)
Fluctuation scaling limits for positive recurrent jumping-in diffusions with small jumps (2020)
Functional limit theorem for occupation time processes of intermittent maps (2020)
Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations (2019)
Approximation and duality problems of refracted processes (2019)
Generalized refracted Levy process and its application to exit problem (2019)
Convergence of Brownian motions on metric measure spaces under Riemannian Curvature–Dimension conditions (2019)
Convergence of continuous stochastic processes on compact metric spaces converging in the Lipschitz distance (2018)
On optimal periodic dividend and capital injection strategies for spectrally negative Levy models (2018)
On optimal periodic dividend strategies for Levy risk processes (2018)
Convergences and projection Markov property of Markov processes on ultrametric spaces (2014)
Strong solutions of Tsirel'son's equation in discrete time taking values in compact spaces with semigroup action (2013)
On a zero-one law for the norm process of transient random walk (2011)
Scaling limit of d-inverse of Brownian motion with functional drift (2010)
Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups (2010)