A Stochastic algorithm to solve multiple dimensional Fredholm integral equations of the second kind

In the present work‎, ‎a new stochastic algorithm is proposed to solve multiple dimensional Fredholm integral equations of the second kind‎. ‎The solution of the‎ integral equation is described by the Neumann series expansion‎. ‎Each term of this expansion can be considered as an expectation which is approximated by a continuous Markov chain Monte Carlo method‎. ‎An algorithm is proposed to simulate a continuous Markov chain with probability density function arisen from an importance sampling technique‎. ‎Theoretical results are established in a normed space to justify the convergence of the proposed method‎. ‎The method has a simple structure and it is a good candidate for parallelization because of the fact that many independent sample paths are used to estimate the solution‎. ‎Numerical results are performed in order to confirm the efficiency and accuracy of the present work‎.