Let $(X_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$ and $\tau$ be a finite stopping time adapted to $(X_t)_{t\ge0}$. Say I run the following algorithm:
Say my abort criterion ensures that $\sum_{i=0}^{n-1}\tau_i$ is a sample from $\tau$. Is this enough to show $$\sum_{i=0}^{n-1}\tau_if(x_i)\approx\operatorname E\left[\int_0^\tau f(X_t)\:{\rm d}t\right]\tag1?$$ Clearly, if $\tau$ is constant and the grid $(\tau_0,\ldots,\tau_{n-1})$ is "fine" enough, $(1)$ is satisfied by the ergodic theorem. But what can we say for random $\tau$? Clearly, the grid still needs to be fine enough, but can we express the error of the approximation in $(1)$ in terms of that grid size maybe?