It is well known from the work of Schönbucher that the marginal laws of a loss process can be matched by a unit increasing time inhomogeneous Markov process, whose deterministic jump intensity is called local intensity. The stochastic local intensity (SLI) models such as the one proposed by Arnsdorf and Halperin allow to get a stochastic jump intensity while keeping the same marginal laws. These models involve a nonlinear stochastic differential equation (SDE) with jumps. The first contribution of this paper is to prove the existence and uniqueness of such processes. This is made by means of an interacting particle system, whose convergence rate toward the nonlinear SDE is analyzed. Second, this approach provides a powerful way to compute pathwise expectations with the SLI model: we show that the computational cost is roughly the same as a crude Monte Carlo algorithm for standard SDEs.