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.

• D. Williams. Probability with Martingales. Cambridge University Press.

• J. Jacod and Ph. E. Protter. Probability Essentials. Springer. 

• E. Çinlar. Probability and Stochastics. Springer.

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

• Cluster construction and limit properties of renewal Hawkes processes (2023)

• 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

• The asymptotic behavior of the renormalized zero resolvent of Lévy processes under regular variation conditions (2025)

• 到達時刻の長時間漸近挙動と到達地点の極限分布 (2025)

• 一般化されたディリクレ問題について (2025)

• Two-point local time penalizations with various clocks for Levy processes (2024)

• 無限過去を持つ作用発展に対する情報系の分解 (2024)

• 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

• Two-point local time penalizations with various clocks for Lévy processes (2025)

• Arcsine and Darling--Kac laws for piecewise linear random interval maps (2023)

• 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)