Randomised stopping time

Question

At the beginning of Ferguson's notes on Optimal Stopping (https://www.math.ucla.edu/~tom/Stopping/Contents.html) a general definition of stopping rule is introduced. Denoting with $N$ the random time at which stopping occurs, associated with the observed process $\{X_n\}$, it is defined in terms of its probability of stopping at time $n$, given observations $X_1=x_1,\ldots,X_n=x_n$, that is in terms of the functions $\phi_n(x_1,\ldots,x_n)=\mathbb{P}(N=n|N\geq n,X_1=x_1,\ldots,X_n=x_n)$, where $0\leq\phi_n\leq 1$. The author calls this a randomised stopping rule. According to this definition, a randomised stopping rule is not in general a stopping time, since the event of stopping is not determined by the process $X$ alone (as usual, I call a stopping time a random time $N$ such that $\{N=n\}\in\mathcal{F}_n=\sigma(X_1,\ldots,X_n)$). Is my understanding correct? Thanks for sharing any advice. I found several definitions of randomised stopping rules on the internet, but there seems to be little agreement, and this seems to be the most general I encountered so far.

Answer 1

A randomized stopping rule corresponds to a stopping time for an enlarged filtration. Let $U_1,U_2,\dots$ be i.i.d. variables that are uniform in $[0,1]$ and independent of the process $\{X_j\}_{j \ge 1}$. Let $\mathcal G_n$ be the $\sigma$-field generated by $X_1,U_1,\dots,X_n,U_n$. Then the stopping time $\tau$ (I prefer that notation rather than $N$) can be defined by $$\tau:=\min\{n \ge 1 :\, U_n\le \phi_n(X_1,\dots,X_n)\}\,.$$ Clearly, $\tau$ is a stopping time for the filtration $\{\mathcal G_n\}$, and it satisfies $$\mathbb{P}(\tau=n|\tau\geq n,X_1=x_1,\ldots,X_n=x_n) =\phi_n(x_1,\ldots,x_n) \,.$$