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

随时间演变的随机现象通过随机过程进行数学建模。概率符号和导出的指标用于分析模型的某些性质是否与真实的世界现象相吻合。到目前为止发展的理论仅限于时间齐次的情况。这导致了随机建模领域的问题，因为温度或资产价格不会以时间均匀的方式演变。数理金融学和地质学中的一些模型考虑了时间非齐性，不能用符号和指数进行分析。为了能够以一种简单而优雅的方式来分析这样一个更大的过程类，我们想推广各自的概念和结果。结果可用于物理学（路径的Hausdorff维数），数值（随机微分方程解的近似）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).