I know how to do the math on this, but I'm running up against not knowing if there is a standard or even a common way to express it.
Let $B(t)$, $C(t)$, $D(t)$, etc. be functions in time. Each of these can be considered to be a function in itself, or a sample of a stochastic process: $\left \lbrace B(t) \right\rbrace$, $\left \lbrace C(t) \right\rbrace$, $\left \lbrace D(t) \right\rbrace$, etc.
Let $t_A \left(B(t), C(t), D(t), \cdots \right)$ be a deterministic function of $B(t)$, $C(t)$, $D(t)$, etc.
I want to express the probability that $t_A < \tau$ for some mix $B, C, D$ being deterministic or stochastic.
At this point, the best I can come up with is, for example, if I have fixed $B(t)$, $C(t)$, but $t_A$ depends on $\left \lbrace D(t) \right\rbrace$ as a stochastic process, then I'd say $$P\left( t_A < \tau \left | B(t), C(t) \right . \right)$$ and either state in the surrounding text, or leave it up to the reader to infer from prior discussion that $P\left( t_A < \tau \left | B(t), C(t) \right . \right)$ depends on the properties of $\left \lbrace D(t) \right\rbrace$.
By extension, $P\left( t_A < \tau \left | C(t) \right . \right)$ would imply that both $\left \lbrace B(t) \right\rbrace$, $\left \lbrace D(t) \right\rbrace$ are stochastic -- hopefully the other combinations would be obvious.
Is that more or less the standard notation? Or is there some clear and common usage that I've missed?
(For background: I'm working on a complicated problem where we'll be simulating things in stages and caching results: so we may simulate against a bunch of instances of $\left \lbrace B(t) \right\rbrace$, then use each instance of $B(t)$ to generate simulation results against a bunch of instances of $\left \lbrace C(t) \right\rbrace$, etc.). Again, I know how to do it, and how to do the math -- I just don't want to reinvent a notational wheel.
To me, this looks like pretty standard random variable notation. I am not familiar with any 'special' notation used for dealing with with your specific problem.
Your $t_A$ is literally a mapping from the sample space to $\mathbf{R}$, so it is a random variable.
And we can condition on the value a random variable takes on. Sometimes, that is expressed as $P(T < t| Y = y)$, for example. But I don't think the $Y = y$ really adds anything other than explicitly turning the condition into an event.