Let (omega,F,P) be a probability space. For each G in F, define G as the s-field generated by G and those sets f in F satisfying P(f) in {0, 1}. Conditions for P to be atomic on the intersection of the complements of Ai for i=1,..,k, with A1, . . . ,Ak in F sub-s-fields, are given. Conditions...

The problem of finding the best-possible lower bound on the distribution of a non-decreasing function of n dependent risks is solved when n=2 and a lower bound on the copula of the portfolio is provided. The problem gets much more complicated in arbitrary dimensions. When no information on the...

We theoretically compare variances between the Infinitesimal Perturbation Analysis (IPA) estimator and the Likelihood Ratio (LR) estimator to Monte Carlo gradient for stochastic systems. The conditions proposed in [Cui et al., 2020] when the IPA estimator has a smaller variance can yield sharper...