  We study the optimal multiple stopping time problem defined for each stopping
time $S$ by $v(S)=\operatorname {ess}\sup_{\tau_1,...,\tau_d\geq
S}E[\psi(\tau_1,...,\tau_d)|\mathcal{F}_S]$. The key point is the construction
of a new reward $\phi$ such that the value function $v(S)$ also satisfies
$v(S)=\operatorname {ess}\sup_{\theta\geq S}E[\phi(\theta)|\mathcal{F}_S]$.
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 for 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 new applications,
particularly in finance (applications to American options with multiple
exercise times).
