非局部单侧伪微分算子的边界条件及其随机过程II
Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes II
摘要：
我们将单侧伪微分算子的边界条件与修正的单侧Lévy过程的生成元联系起来。一方面，这使得建模者在限制建模域时可以放心地使用适当的边界条件。另一方面，它允许基于微分方程求解器的数值技术，以获得密度的快速近似值或在金融等领域遇到的受限单边Lévy过程的其他统计特性。特别是，我们确定了一个新的非局部质量守恒边界条件，显示它对应于快进，即删除的时间外的域的过程。我们处理所有的组合杀死，反射和快进边界条件。在第一部分中，我们证明了带边界条件的单侧伪微分算子作为生成元的向后和向前柯西问题的适定性。我们这样做的基础上的修改Lévy过程的网格点近似的Feller半群的收敛性。在第二部分中，我们证明了极限Feller半群确实是与修改的Lévy过程相关联的半群，通过展示关于Skorokhod拓扑的修改的连续性。
英文摘要：
We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided Lévy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelling domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided Lévy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corresponds to fast-forwarding, i.e. removing the time the process spends outside the domain. We treat all combinations of killing, reflecting and fast-forwarding boundary conditions. In Part I we show wellposedness of the backward and forward Cauchy problems with a one-sided pseudo-differential operator with boundary conditions as generator. We do so by showing convergence of Feller semigroups based on grid point approximations of the modified Lévy process. In Part II we show that the limiting Feller semigroup is indeed the semigroup associated with the modified Lévy process by showing continuity of the modifications with respect to the Skorokhod topology.
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