论文标题
一般路径积分和稳定的SDE
General path integrals and stable SDEs
论文作者
论文摘要
由布朗运动驱动的一维随机微分方程的理论是经典的，并且在很大程度上被理解了几十年。对于带有跳跃的随机微分方程，图片仍然不完整，即使是一些最基本的问题也只能部分理解。在本文中，我们研究了\ [ {\ rm d} z_t =σ（z_ {t - }）{\ rm d} x_t \]以经典的Engelbert-Schmidt时间更改方法为基于（对称）$α$稳定的莱维过程。扩展Zanzotto的结果和完成结果，我们为$α\ in（0,1）$的弱解决方案的存在而得出了一个完整的表征。我们的方法不是基于经典的随机演算论证，而是基于马尔可夫过程的一般理论。我们证明积分测试是在最小假设下的路径积分的有限性。
The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to \[ {\rm d}Z_t=σ(Z_{t-}){\rm d} X_t \]driven by a (symmetric) $α$-stable Lévy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for $α\in(0,1)$. Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We proof integral tests for finiteness of path integrals under minimal assumptions.
