Optimal multiple stopping time problem
Optimal multiple stopping time problem
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DOI：
10.1214/10-aap727
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发表时间：
2009-10
影响因子：
1.8
通讯作者：
M. Kobylanski;M. Quenez;Elisabeth Rouy-Mironescu
中科院分区：
文献类型：
--
作者：
M. Kobylanski;M. Quenez;Elisabeth Rouy-Mironescu
We study the optimal multiple stopping time problem defined for each stopping time $S$ by $\displaystyle{ v(S)=\esssup_ {\tau_1,\cdots,\tau_d \geq S } E[\psi( \tau_1,\cdots,\tau_d)\, |\,\F_S]\,}$.\\ The key point is the construction of a {\em new reward} $\phi$ such that the value function $v(S)$ satisfies also $ v(S)=\esssup_ {\theta \geq S } \,E[\phi(\theta)\, |\,\F_S]\,.$ This new reward $\phi$ is not a right continuous adapted process as in the classical case but a family of random variables. For such a reward, we prove a new existence result of optimal stopping times under weaker assumptions than in the classical case. This result is used to prove the existence of optimal multiple stopping times for $v(S)$ by a constructive method. Moreover, under strong regularity assumptions on $\psi$, we show that the new reward $\phi$ can be aggregated by a progressive process. This leads to different applications in particular in finance for American options with multiple exercise times.
