Cylindrical Levy Processes and Their Applications

随机微分方程是一个随时间演化并受随机噪声影响的过程的模型。自然界和经济学中的许多现象都可以用这些方程来建模。产生随机噪声的原因可以在外部或内部波动中找到，这些波动不允许进行确定性描述，也可以在未来的随机事件或模型的不确定性中找到。模型的复杂性，例如所涉及的参数数量或建模过程的状态空间，常常导致需要考虑无限维空间中的随机微分方程。然而，到目前为止，由于缺乏令人满意的数学理论，这些模型大多局限于连续的高斯噪声和结构非常丰富的无限维空间。这个项目的第一个目标是发展一种理论，使我们能够处理一般类型的无限维空间中的随机微分方程。随机源可能具有不连续的路径，并且允许具有非常一般的形式，使得随机性不仅取决于时间的演变，而且取决于底层空间。在该项目的第二部分，通过研究两个具体的应用实例来验证该理论的可用性：一个模型描述了受某些外部随机噪声影响的给定区域的热量物理分布，第二个模型源于金融数学，描述了利率曲线的演变。

Stochastic differential equations model a process evolving in time and subject to a random noise. Numerous phenomena in nature and economics are modelled by these equations. The reason for the random noise might be found in external or internal fluctuations which do not allow a deterministic description, in random events in the future or in uncertainty of the model. The complexity of the model, e.g. the numbers of parameters involved or the state space of the modelled process, often results in the necessity to consider stochastic differential equations in infinite dimensional spaces. However, up to now, most of these models are restricted to a continuous Gaussian noise and to infinite dimensional spaces with a very rich structure due to the lack of a satisfactory mathematical theory.The first objective of this project is to develop a theory which enables us to treat stochastic differential equations in infinite dimensional spaces of a general type. The random source might have discontinuous paths and is allowed to be of a very general form, such that the randomness not only depends on the evolution in time but also on the underlying space. In the second part of this project, the usability of the theory is verified by studying two concrete examples out of the numerous applications: one model describes the physical distribution of the heat in a given region subject to some external random noise, and the second model originates from financial mathematics and describes the evolution of interest rate curves.