  It is well-known from the work of Sch\"onbucher (2005) 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 (2008) allow to get a stochastic jump intensity while keeping the
same marginal laws. These models involve a non-linear 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 towards the non-linear 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.
