论文标题
非局部单方面伪差异算子和相关随机过程I的边界条件I
Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes I
论文作者
论文摘要
我们将单方面伪差异运算符的边界条件与修改后的Lévy过程的发生器联系起来。一方面，这允许建模者在限制建模域时自信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获得遇到的限制性单方面莱维过程的密度或其他统计特性的快速近似，例如在金融中。特别是，我们通过表明其对应于快进，即删除该过程在域之外的时间来确定新的非局部质量保护边界条件。我们处理杀戮，反射和快速朝向边界条件的所有组合。 在第一部分中，我们显示出具有边界条件作为生成器的单方面伪差异操作员的向后和前进问题的良好性。我们通过根据修改后的Lévy过程的网格点近似来显示Feller Semigroups的收敛性。 在第二部分中，我们表明，限制的Feller Semigroup确实是通过显示有关Skorokhod拓扑的修改的连续性，与修改后的Lévy过程相关的半群。
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.
