Data-driven satisficing measure and ranking

We propose an computational framework for real-time risk assessment and prioritizing for random outcomes without prior information on probability distributions. The basic model is built based on satisficing measure (SM) which yields a single index for risk comparison. Since SM is a dual representation for a family of risk measures, we consider problems constrained by general convex risk measures and specifically by Conditional value-at-risk. Starting from offline optimization, we apply sample average approximation technique and argue the convergence rate and validation of optimal solutions. In online stochastic optimization case, we develop primal-dual stochastic approximation algorithms respectively for general risk constrained problems, and derive their regret bounds. For both offline and online cases, we illustrate the relationship between risk ranking accuracy with sample size (or iterations).Comment: 26 Pages, 6 Figure

On the rate of convergence to stationarity of the M/M/N queue in the Halfin-Whitt regime

We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the M/M/n queue in the Halfin-Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic phase transition that occurs w.r.t. this rate. In particular, we demonstrate the existence of a constant $B^*\approx1.85772$ s.t. when a certain excess parameter $B\in(0,B^*]$, the error in the steady-state approximation converges exponentially fast to zero at rate $\frac{B^2}{4}$. For $B>B^*$, the error in the steady-state approximation converges exponentially fast to zero at a different rate, which is the solution to an explicit equation given in terms of special functions. This result may be interpreted as an asymptotic version of a phase transition proven to occur for any fixed n by van Doorn [Stochastic Monotonicity and Queueing Applications of Birth-death Processes (1981) Springer]. We also prove explicit bounds on the distance to stationarity for the M/M/n queue in the Halfin-Whitt regime, when $B<B^*$. Our bounds scale independently of $n$ in the Halfin-Whitt regime, and do not follow from the weak-convergence theory.Comment: Published in at http://dx.doi.org/10.1214/12-AAP889 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A strongly convergent numerical scheme from Ensemble Kalman inversion

The Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman inversion. The results of this paper make a step towards closing the gap of the missing approximation result by proving a strong convergence result in a simplified model of a scalar stochastic differential equation. We focus here on a toy model with similar properties than the one arising in the context of Ensemble Kalman filter. The proposed model can be interpreted as a single particle filter for a linear map and thus forms the basis for further analysis. The difficulty in the analysis arises from the formally derived limiting SDE with non-globally Lipschitz continuous nonlinearities both in the drift and in the diffusion. Here the standard Euler-Maruyama scheme might fail to provide a strongly convergent numerical scheme and taming is necessary. In contrast to the strong taming usually used, the method presented here provides a weaker form of taming. We present a strong convergence analysis by first proving convergence on a domain of high probability by using a cut-off or localisation, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence

Weak Convergence in the Prokhorov Metric of Methods for Stochastic Differential Equations

We consider the weak convergence of numerical methods for stochastic differential equations (SDEs). Weak convergence is usually expressed in terms of the convergence of expected values of test functions of the trajectories. Here we present an alternative formulation of weak convergence in terms of the well-known Prokhorov metric on spaces of random variables. For a general class of methods, we establish bounds on the rates of convergence in terms of the Prokhorov metric. In doing so, we revisit the original proofs of weak convergence and show explicitly how the bounds on the error depend on the smoothness of the test functions. As an application of our result, we use the Strassen - Dudley theorem to show that the numerical approximation and the true solution to the system of SDEs can be re-embedded in a probability space in such a way that the method converges there in a strong sense. One corollary of this last result is that the method converges in the Wasserstein distance, another metric on spaces of random variables. Another corollary establishes rates of convergence for expected values of test functions assuming only local Lipschitz continuity. We conclude with a review of the existing results for pathwise convergence of weakly converging methods and the corresponding strong results available under re-embedding.Comment: 12 pages, 2nd revision for IMA J Numerical Analysis. Further minor errors correcte