An unbiased Monte Carlo method to solve linear Volterra equations of the second kind

In previous works (Dimov and Maire in Adv Comput Math 45(3):1499–1519, 2019; Dimov et al. in Appl Math Model 39(15):4494–4510, https://doi.org/10.1016/j.apm.2014.12.018 , 2015), we have developed two Monte Carlo algorithms to solve linear systems and Fredholm integral equations of the second kind. These algorithms rely on the computation of a score along a discrete or continuous homogeneous Markov chain until absorption. Here, we propose two approaches to extend the Fredholm algorithm to Volterra equations. The first one is based on a change in variable at each step of the Markov chain. The second one uses the indicator function to transform the Volterra equation into an appropriate form. The resulting Markov chains are inhomogeneous with an increasing absorption rate. The convergence is ensured as soon as the Volterra kernel is bounded. Numerical examples are given on basic reference problems and on high dimensional test cases up to 100 dimensions.