Numerical methods for quasi-stationary distributions

In stochastic processes with absorbing states, the quasi-stationary distribution provides valuable insights into the long-term behavior prior to absorption. In this work, we revisit two well-established numerical methods for its computation. The first is an iterative algorithm for solving the nonlinear equation that defines the quasi-stationary distribution. We generalize this technique to accommodate general Markov stochastic processes, either with discrete or continuous state space, and with multiple absorbing states. The second is a Monte Carlo method with resetting, for which we propose a single-trajectory approach that uses the trajectory's own history to perform resets after absorption. In addition to these methodological contributions, we provide a detailed analysis of implementation aspects for both methods. We also compare their accuracy and efficiency across a range of examples. The results indicate that the iterative algorithm is generally the preferred choice for problems with simple boundaries, while the Monte Carlo approach is more suitable for problems with complex boundaries, where the implementation of the iterative algorithm is a challenging task.