The following paper by Thomas Galtier et al. gives the probability density function (pdf) of a PDMP trajectory of fixed duration and its associated reference measure : www.esaim-ps.org/articles/ps/... 12/n

The idea is to generate the PDMP trajectories according to an alternative (importance) distribution that realizes the event of interest more frequently, and to rectify the bias of the estimator by introducing a suited likelihood ratio. 11/n

A standard Monte Carlo method (generating a large number of PDMP trajectories modeling the system, and returning the proportion of simulations realizing the event of interest) is inefficient when the event is rare. We therefore turn to an importance sampling method. 10/n

Critical failure occurs when physical variables exceed a given threshold, and is only possible when some key combinations of components fail at the same time and are not repaired in time. It is therefore a rare event (of the order of 10^-5 at most in our test cases). 9/n

The distribution of waiting times before breakdown or repair depends on the state of the components and continuously on the physical variables. We are interested in the probability that the system will go into critical failure before a fixed date. 8/n

The system's continuous variables (temperature, pressure, liquid levels, etc.) evolve according to deterministic physical laws that depend on the state of the various system components (pumps, valves, generators, etc.), which are subject to random breakdowns and repairs. 7/n

PDMPs model a wide range of phenomena, and are particularly popular in biology. They also form the basis of PD-MCMC methods in Bayesian inference for simulating high-dimensional distributions. We use them to model complex hybrid dynamic industrial systems. 6/n

A PDMP is a "hybrid" continuous-time stochastic process. It jumps at random times to new states like a classical non-diffusive Markovian process, but follows deterministic (non-trivial) trajectories between jumps. 5/n

I give some details below on PDMPs, rare event simulation, importance sampling, the committor function, the cross entropy method, recycling schemes and our family of approximations of the committor function. 4/n

Our contribution: a new family of importance distributions based on committor function approximation for PDMPs modeling complex high-dimensional industrial systems, and a central limit theorem for the adaptive importance sampling procedure of PDMPs with recycling scheme. 3/n

Our goal: to estimate the (very small) probability that a PDMP (piecewise deterministic Markov process) trajectory will reach a critical region of its state space before a fixed date, using an importance sampling method. 2/n

Some keywords not in the title of the paper: rare event simulation, reliability assessment, cross entropy minimization with recycling scheme, Monte Carlo, stochastic processes, committor function approximation. 1/n