@Nate Yes, the angle brackets are used as parenthesis for the expectation. I'll change them to square brackets. I see now how $X$ needs to be specified a more. I am looking into hitting times for SDEs, in particular the Ornstein-Uhlenbeck $dY(x) = (ax+b)dx + \sigma dW(x)$ because it has an explicit solution. The random variable X is defined as the values x for which Y(x)=y. First exit times can be studied via the Backward Kolmogorov with specified region boundaries, but I was wondering if some information about hitting times (not "first" anymore) can be obtained directly from the SDE solution.