Monte Carlo (MC) is widely used for the simulation of discrete time Markov chains. We consider the case of a d-dimensional continuous state space and we restrict ourselves to chains where the d components are advanced independently from each other, with d random numbers used at each step. We simulate N copies of the chain in parallel, and we replace pseudorandom numbers on I^d := (0,1)^d with stratified random points over I^2d : for each point, the first d components are used to select a state and the last d components are used to advance the chain by one step. We use a simple stratification technique: let p be an integer, then for N=p^2d samples, the unit hypercube is dissected into N hypercubes of measure 1/N and there is one sample in each of them. The strategy outperforms classical MC if a well-chosen multivariate sort of the states is employed to order the chains at each step. We prove that the variance of the stratified sampling estimator is bounded by 𝒪(N^-(1+1/(2d))) , while it is 𝒪(N^-1) for MC. In numerical experiments, we observe empirical rates that satisfy the bounds. We also compare with the Array-RQMC method.
This paper provides a framework in which multilevel Monte Carlo and continuous level Monte Carlo can be compared. In continuous level Monte Carlo the level of refinement is determined by an exponentially distributed random variable, which therefore heavily influences the computational complexity. We propose in this paper a variant of the algorithm, where the exponentially distributed random variable is generated by a quasi Monte Carlo sequence, resulting in a significant variance reduction. In the examples presented the quasi continuous level Monte Carlo algorithm outperforms multilevel and continuous level Monte Carlo by a clear margin.
The performance of stochastic numerical methods for bounded diffusions (accuracy, speed and weak rate of convergence) is considered. The backdrop is the pointwise solution, via stochastic representations, of boundary value problems with mixed boundary conditions. Three stochastic solvers are tested on application-inspired problems, and their performance is noticeably improved thanks to simple heuristics introduced here.
We analyze the ruin event in a Markovian insurance risk model. For actual computations of risk functionals, we sketch different numerical approaches and focus on assessing the performance of a quantization algorithm. Since by nature ruin should be a rare event, it is necessary to deploy a variance reduction technique based on a proper change of measure.
We discuss a generalisation of the multilevel Markov Chain Monte Carlo algorithm for a Bayesian inverse problem with high-resolution (full-field) data. We extend the method to include a level-dependent treatment of the data, which is useful for very high-resolution data that cannot be represented by the much lower resolution of the forward problem at coarser levels. The approach is illustrated using a model case situated in a context of structural health monitoring, additionally providing a new application domain for the multilevel Markov Chain Monte Carlo methodology.
