Volume 15, Issue 3

Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) are two risk measures which are widely used in the practice of risk management. This paper deals with the problem of estimating both VaR and CVaR using stochastic approximation (with decreasing steps): we propose a first Robbins–Monro (RM) procedure based on Rockafellar–Uryasev's identity for the CVaR. Convergence rate of this algorithm to its target satisfies a Gaussian Central Limit Theorem. As a second step, in order to speed up the initial procedure, we propose a recursive and adaptive importance sampling (IS) procedure which induces a significant variance reduction of both VaR and CVaR procedures. This idea, which has been investigated by many authors, follows a new approach introduced in [Lemaire and Pagès, Unconstrained Recursive Importance Sampling, 2008]. Finally, to speed up the initialization phase of the IS algorithm, we replace the original confidence level of the VaR by a slowly moving risk level. We prove that the weak convergence rate of the resulting procedure is ruled by a Central Limit Theorem with minimal variance and its efficiency is illustrated on several typical energy portfolios.

This work produces a statistical description of several sample-based techniques for numerical integration probabilistic assessment. The sampling techniques include pseudo-random, quasi-random, and centroidal Voronoi tessellation methods. The compared statistics are the star-discrepancy, minimum distance between points, and independence of sample sets. These statistics are selected because they influence the size of confidence intervals generated by repeated analyses. The optimal distance between points for two, three and four dimensions is derived as well as the general form for m dimensions. The results show that for problems with few random variables the complex methods would produce smaller confidence intervals. However, the benefits of more complex techniques are marginalized when there are more random variables.

For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, the paper compares the bounds on the rates of convergence to normality (Berry–Esseen type) of two approximate maximum likelihood estimators of the drift parameter based on the Itô and the Fisk–Stratonovich approximations of the continuous likelihood, under some regularity conditions, when the diffusion is observed at equally spaced dense time points over a long time interval. It shows that the Fisk–Stratonovich approximations performs better than the Itô approximations in the sense of having smaller variance.

The Impartial Anonymous and Neutral Culture (IANC) model of social choice assumes that the names of the voters as well as the identity of the alternatives are immaterial. This models allows for comparison of structural properties social choice rules (SCRs) for large values of the parameters empirically: whether Condorcet winners exist, whether Borda and Condorcet winners are identical, whether Plurality with run-off winners are among Borda winners, for example. We derive an exact formula for the number of equivalence classes of preference profiles (called roots) in this model. The number of terms in the formula depends only on the number of alternatives m , and not the much larger number of voters n . In IANC, the equivalence classes defining the roots do not have the same size, making their uniform generation for large values of the parameters nontrivial. We show that the Dixon–Wilf algorithm can be adapted to this problem, and describe a symbolic algebra routine that can be used for Monte-Carlo algorithms for the study of various structural properties of SCRs.

Sparsified Randomization Monte Carlo (SRMC) algorithms for solving large systems of linear algebraic equations are presented. We construct efficient stochastic algorithms based on a probabilistic sampling of small size sub-matrices, or a randomized evaluation of a matrix-vector product and matrix iterations via a random sparsification of the matrix. This approach is beyond the standard Markov chain based Neumann–Ulam method which has no universal instrument to decrease the variance. Instead, in the new method, first, the variance can be decreased by increasing the number of the sampled columns of the matrix in play, and second, it is free of the restricted assumption of the Neumann–Ulam scheme that the Neumann series converges. We apply the developed methods to different stochastic iterative procedures. Application to boundary integral equation of the electrostatic potential theory is given where we develop a SRMC algorithm for solving the approximated system of linear algebraic equations, and compare it with the standard Random Walk on Boundary method.