General path integrals and stable SDEs
General path integrals and stable SDEs
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DOI：
10.4171/jems/1331
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发表时间：
2020-12
影响因子：
2.6
通讯作者：
S. Baguley;L. Doering;A. Kyprianou
中科院分区：
文献类型：
--
作者：
S. Baguley;L. Doering;A. Kyprianou
由布朗运动驱动的一维随机微分方程理论是经典的，几十年来已被广泛理解。对于带有跳跃的随机微分方程，图片仍然不完整，甚至一些最基本的问题也只是部分理解。在本文中，我们本着经典 Engelbert-Schmidt 时间变化方法的精神，研究由（对称）$\alpha$-稳定 Levy 过程驱动的 \[ {\rm d}Z_t=\sigma(Z_{t-}){\rm d} X_t \] 弱解的存在性和唯一性。扩展和完善 Zanzotto 的结果，我们得出了 $\alpha\in(0,1)$ 弱解的存在性和唯一性的完整表征。我们的方法不是基于经典的随机微积分论证，而是基于马尔可夫过程的一般理论。我们在最小假设下证明路径积分的有限性的积分测试。
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 \[ {\rm d}Z_t=\sigma(Z_{t-}){\rm d} X_t \]driven by a (symmetric) $\alpha$-stable Levy 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 $\alpha\in(0,1)$. 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.