An efficient Multilevel Splitting scheme 1

Rare event analysis has been attracting continuous and growing attention over the 4 past decades. It has many possible applications in different areas, e.g., queueing 5 theory, insurance, engineering etc. As explicit expressions are hard to obtain, and 6 asymptotic approximations often lack error bounds, one often applies simulation 7 methods to obtain performance measures of interest. 8 Obviously, the use of standard Monte Carlo simulation for estimating rare event 9 probabilities has an inherent problem: it is extremely time consuming to obtain 10 reliable estimates since the number of samples needed to obtain an estimate of a 11 certain predefined accuracy is inversely proportional to the probability of interest. 12 Two important techniques to speed up simulations are Importance Sampling (IS) 13 and Multilevel Splitting (MS). 14 IS prescribes to simulate the system under a new probability measure such that 15 the event of interest occurs more frequently, and corrects the simulation output by 16 means of likelihood ratios to retain unbiasedness. The likelihood ratios essentially 17 capture the likelihood of the realization under the old measure with respect to the 18 new measure. The choice of a ‘good’ new measure is rather delicate; in fact only 19 measures that are asymptotically efficient are worthwile to consider. We refer to 20 [3] for more background on IS and its pitfalls. 21 The other technique, multilevel splitting (MS), is conceptually easier, in the 22 sense that one can simulate under the normal probability measure. When a sample 23 path of the process is simulated, this is viewed as the path of a ‘particle’. When 24 the particle approaches the target set to a certain distance, the particle splits 25 into a number of new particles, each of which is then simulated independently of 26 each other. This process may repeat itself several times, hence the term multilevel 27 splitting. Typically, the states where particles should be split are determined by 28 selecting a number of level sets of an importance function f . Every time a particle 29 (sample path) crosses the next level set of the importance function f , it is split. 30 The splitting factor (i.e. the number of particles that replaces the original particle) 31 may depend on the current level. 32