Analysis of path properties of stochastic processes via generalized indices under allowance of time inhomogeneity

通过随机过程对随时间演化的随机现象进行数学建模。使用概率符号和导出的指数来分析模型的某些性质是否符合真实世界现象。到目前为止，所发展的理论仅限于时间均匀的情况。这导致了随机建模领域的问题，因为温度或资产价格不是以时间均匀的方式演变的。数学、金融和地质学中的一些允许时间不均匀的模型，不能用符号和指数来分析。为了能够以一种简单而优雅的方式来分析如此大的一类过程，我们想要概括各自的概念和结果。这些结果可用于物理(路径的Hausdorff维)、数值(随机微分方程解的逼近)和数学金融(路径的变化)。

Random phenomena which evolve in time are modeled mathematically via stochastic processes. The probabilistic symbol and the derived indices are used to analyze whether certain properties of the model coincide with the real world phenomenon. The theory developed until now is restricted to the time homogeneous case. This results in problems in the area of stochastic modeling, since temperature or asset prices do not evolve in a time homogeneous manner. Some models in mathematical finance and geology which allow for time inhomogeneity, cannot be analyzed by using the symbol and the indices. In order to allow for an easy and elegant way of analyzing such a bigger class of processes, we want to generalize the respective concepts and results. The results can be used in physics (Hausdorff dimension of the paths), numerics (approximation of solutions of stochastic differential equations) and mathematical finance (variation of the paths).