On the properties of $r$-excessive mappings for a class of diffusions
On the properties of $r$-excessive mappings for a class of diffusions
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关于一类扩散的$r$-过度映射的性质
DOI：
10.1214/aoap/1069786509
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发表时间：
2003
影响因子：
1.8
通讯作者：
L. Alvarez
中科院分区：
文献类型：
--
作者：
L. Alvarez
对于给定的线性、时齐和正则扩散过程，我们考虑了一类r-调和映象的凸性和相对静态性质。给出了所考虑扩散的极小r-过度映射是凸的一组弱条件，以及任意非平凡r-过度映射在r-调和区域上是凸的条件。因此，我们能够给出一组通常满足的条件，在这些条件下，增加的波动性增加了r-调和映射值。我们将我们的结果应用于研究永久美式未定权益定价中经常出现的一类最优停止问题，并给出了价值函数在连续区域上凸的一组条件，在此条件下，波动率的增加明确地增加了价值函数，扩大了连续区域，从而推迟了索赔的合理行使。
We consider the convexity and comparative static properties of a class of r-harmonic mappings for a given linear, time-homogeneous and regular diffusion process. We present a set of weak conditions under which the minimal r-excessive mappings for the considered diffusion are convex and under which an arbitrary nontrivial r-excessive mapping is convex on the regions where it is r-harmonic. Consequently, we are able to present a set of usually satisfied conditions under which increased volatility increases the value of r-harmonic mappings. We apply our results to a class of optimal stopping problems arising frequently in studies considering the pricing of perpetual American contingent claims and state a set of conditions under which the value function is convex on the continuation region and, consequently, under which increased volatility unambiguously increases the value function and expands the continuation region, thus postponing the rational exercise of the claim.