Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping
Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping
复制标题
本地时间的路径不等式：在 Skorokhod 嵌入和最优停止中的应用
DOI：
--
复制
发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Jan Obl'oj
中科院分区：
文献类型：
--
作者：
A. Cox;D. Hobson;Jan Obl'oj
本文建立了一类形式为$H（B t）\geM_t+F（L_t）$的路径不等式，其中$B_t$是布朗运动，$L_t$是其在零点的局部时，$M_t$是局部鞅.该表示的具体性质使不等式适用于各种应用。在这项工作中，我们使用的不等式得到的Vallois' Skorokhod嵌入的结构和最优性结果。我们讨论他们的金融解释的背景下，强大的定价和对冲期权写在当地时间。在本文的最后一部分，我们利用这些不等式解决了一类最优停止问题$\sup_{\tau}\mathbb{E}[F（L_{\tau}）-\int _0^{\tau}\beta（B_s）ds]$。该解决方案是通过一个微分方程系统的最小解给出的，因此类似于佩斯基尔所描述的极大值原理。在整个过程中，重点放在技术的新奇和简单性上。
We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero and $M_t$ a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois' Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form $\sup_{\tau}\mathbb{E}[F(L_{\tau})-\int _0^{\tau}\beta(B_s) ds]$. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques.