全纯变换及其在仿射过程中的应用
Holomorphic transforms with application to affine processes
摘要：
在Itô-Lévy过程的一个相当一般的设置中，我们研究了一类过程的状态变量的变换（例如傅立叶变换），这些变换在复平面中的时间零点附近的某个圆盘中是全纯的。我们表明，这种变换是有关的系统的分析向量的生成过程中，我们的状态条件，允许全纯扩展这些变换成一个带，其中包含正实轴。基于这些扩展，我们开发了一个功能的系列扩展这些变换的成分的生成器。作为应用，我们证明了在一定的平稳性条件下，具有状态依赖跳跃部分的多维仿射Itô-Lévy过程的傅里叶变换在时间带上是全纯的，并给出了该变换的对数仿射级数表示.
英文摘要：
In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.
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