Open Access

Yusuke Asai

• Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) which have a stochastic process in their vector field functions. RODEs have been used in a wide range of applications such as biology, medicine, population dynamics and engineering and play an important role in the theory of random dynamical systems, however, they have been long overshadowed by stochastic differential equations. Typically, the driving stochastic process has at most Hoelder continuous sample paths and the resulting vector field is, thus, at most Hoelder continuous in time, no matter how smooth the vector function is in its original variables, so the sample paths of the solution are certainly continuously differentiable, but their derivatives are at most Hoelder continuous in time. Consequently, although the classical numerical schemes for ODEs can be applied pathwise to RODEs, they do not achieve their traditional orders. Recently, Gruene and Kloeden derived the explicit averaged Euler scheme by taking the average of the noise within the vector field. In addition, new forms of higher order Taylor-like schemes for RODEs are derived systematically by Jentzen and Kloeden. However, it is still important to build higher order numerical schemes and computationally less expensive schemes as well as numerically stable schemes and this is the motivation of this thesis. The schemes by Gruene and Kloeden and Jentzen and Kloeden are very general, so RODEs with special structure, i.e., RODEs with Ito noise and RODEs with affine structure, are focused and numerical schemes which exploit these special structures are investigated. The developed numerical schemes are applied to several mathematical models in biology and medicine. In order to see the performance of the numerical schemes, trajectories of solutions are illustrated. In addition, the error vs. step sizes as well as the computational costs are compared among newly developed schemes and the schemes in literature.
• Zufällige gewöhnliche Differentialgleichungen (englisch: Random Ordinary Differential Equations, Akronym: RODEs) sind gewöhnliche Differentialgleichungen (englisch: Ordinary Differential Equations, Akronym: ODEs), die einen stochastischen Prozess in ihrer Vektorfeld-Funktion haben. RODEs werden in einer Vielzahl von Anwendungen, z.B. in der Biologie, Medizin, Populationsdynamik und der Technik eingesetzt [15, 70, 81, 90, 92] und spielen eine wichtige Rolle in der Theorie der zufälligen dynamischen Systeme [5]. Lange jedoch standen sie im Schatten von stochastischen Differentialgleichungen (englisch: Stochastic Differential Equations, Akronym: SDEs)...
• ランダム常微分方程式(Random Ordinary Differential Equations，以下，RODEs) は，そのベクトル場をなす関数に確率過程を伴った常微分方 程式(Ordinary Differential Equations，以下，ODEs) である．RODEs は， 生物学，医学，人口動態および工学といった様々な分野[15, 70, 81, 90, 92] で応用され，さらにランダムな力学系の理論において重要な役割を果たし てきたが[5]，長い間，確率微分方程式(Stochastic Differential Equations， 以下，SDEs) の影に隠れる存在であった． 一般に，Rd1 において，RODEs は...