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Having described the general principles and some known versions of splitting, we now discuss several key issues that need to be addressed for an ef cient implementation of splitting First, how should the importance function h be de ned This is de nitely the most important and most dif cult question to address For multilevel splitting, in the simple case where the state space is one-dimensional and included in R, the nal time is an almost surely nite stopping time, and the critical region has the form B = [b, ), then all strictly increasing functions h are equivalent if we assume that we have the freedom to select the levels (it suf ces to move the levels to obtain the same subsets Bk ) So we can just take h(x) = x, for
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instance Otherwise, especially if the state space is multidimensional, the question is much more complicated Indeed, the importance function is a one-dimensional projection of the state space Under simplifying assumptions, it is shown in [16] and below that ideally, to minimize the residual variance of the estimator from the current stage onward, the probability of reaching the next level should be the same at each possible entrance state to the current level This is equivalent to having h(x) proportional to P[TB T | X(0) = x] But if we knew these probabilities, we would know the exact solution and there would be no need for simulation In this sense, this is a similar issue to that of the optimal (zero-variance) change of measure in IS The idea is then to use an approximation of P[TB T | X(0) = x] or an adaptive (learning) technique One way to learn the importance function was proposed in [4]: the state space is partitioned in a nite number of regions and the importance function h is assumed to be constant in each region The average value of P[TB T | X(0) = x] in each region is estimated by the fraction of chains that reach B among those that have entered this region These estimates are combined to de ne the importance function for further simulations, which are used in turn to improve the estimates, and so on We will see in Section 35, on a simple tandem queue, that the choice of the importance function is really a critical issue; an intuitively appealing (but otherwise poor) selection can lead to high inef ciency It is important to emphasize that the above analysis considers only the variance and not the computing time (the work) If we take the work into account (which we should normally do) then taking h(x) proportional to P[TB T | X(0) = x] is not necessarily optimal, because the expected work to reach B may depend substantially on the current state x In a rare event setting, it is important to understand how a proposed importance function would behave asymptotically as a function of the rare event probability when 0, that is, in a rare event asymptotic regime This type of analysis is pursued in [9], in a framework where is assumed to be well approximated by a large-deviation limit, for which the rate of decay is described by the solution of the Hamilton Jacobi Bellman (HJB) nonlinear partial differential equations associated with some control problem The authors show that a good importance function must be a viscosity subsolution of the HJB equations, multiplied by an appropriate scalar selected so that the probability of reaching a given level k from the previous level k 1 is 1/Ok 1 when Lk = k 1 In the context of xed splitting, this condition is necessary and suf cient for the expected total number of particles not to grow exponentially with log Moreover, if the subsolution also has its maximal possible value at a certain point, then the splitting scheme is asymptotically optimal, in the sense that the relative variance grows slower than exponentially in log Second, how should the number of offspring be chosen In xed splitting, the question is how to select the number Ok of offspring at each level If we do not split enough, reaching the next level (and the rare event) becomes unlikely On the other hand, if we split too much, the number of trajectories will explode
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