Kolmogorov's extension theorem is generalized so that the time domain is a random set. This is applied to the construction of stochastic processes with random time domains, generalizing certain results of Dynkin and Kuznetsov.
K. Y. Hu. "A Generalization of Kolmogorov's Extension Theorem and an Application to the Construction of Stochastic Processes with Random Time Domains." Ann. Probab. 16 (1) 222 - 230, January, 1988. https://doi.org/10.1214/aop/1176991896