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Let ρ be a general law-invariant convex risk measure, for instance, the average value at risk, and let X be a financial loss, that is, a real random variable. In practice, either the true distribution μ of X is unknown, or the numerical computation of ρ (μ) is not possible. In both cases, either relying on historical data or using a Monte Carlo approach, one can resort to an independent and identically distributed sample of μ to approximate ρ (μ) by the finite sample estimator ρ (μ N) (μN denotes the empirical measure of μ). In this article, we investigate convergence rates of ρ (μ N) to ρ (μ). We provide nonasymptotic convergence rates for both the deviation probability and the expectation of the estimation error. The sharpness of these convergence rates is analyzed. Our framework further allows for hedging, and the convergence rates we obtain depend on neither the dimension of the underlying assets nor the number of options available for trading. Funding: Daniel Bartl is grateful for financial support through the Vienna Science and Technology Fund [Grant MA16-021] and the Austrian Science Fund [Grants ESP-31 and P34743]. Ludovic Tangpi is supported by the National Science Foundation [Grant DMS-2005832] and CAREER award [Grant DMS-2143861]. [ABSTRACT FROM AUTHOR]

This paper presents a general class of dynamic stochastic optimization problems we refer to as stochastic depletion problems. A number of challenging dynamic optimization problems of practical interest are stochastic depletion problems. Optimal solutions for such problems are difficult to obtain, both from a pragmatic computational perspective as well as from a theoretical perspective. As such, simple heuristics are desirable. We isolate two simple properties that, if satisfied by a problem within this class, guarantee that a myopic policy incurs a performance loss of at most 50% relative to the optimal adaptive control policy for that problem. We are able to verify that these two properties are satisfied for several interesting families of stochastic depletion problems and, as a consequence, we identify computationally efficient approximations to optimal control policies for a number of interesting dynamic stochastic optimization problems. [ABSTRACT FROM AUTHOR]

This paper establishes verifiable conditions ensuring the important notion of metric regularity for general nondifferentiable programming problems in Banach spaces. These conditions are used to obtain Lagrange-Kuhn-Tucker multipliers for minimization problems with infinitely many inequality and equality constraints. [ABSTRACT FROM AUTHOR]

We consider several approximations to the lifetime distribution of general k-out-of-n systems with cold standby, when n becomes large. The approximations are derived from limiting results for N(t), the number of failures in [0, t], when t → ∞. The approximations can be easily evaluated even though the derivation of the limiting result is rather technical, due to the discrete character of N(t). Numerical work shows excellent agreement with the theoretical error structure of the approximations. [ABSTRACT FROM AUTHOR]

A central depot periodically orders or produces new stock and withdraws inventory in order to allocate it among several demand points, each experiencing random demands over a finite planning horizon. This system gives rise to a dynamic program with a state space of very large dimension. We determine a near-optimal allocation policy and construct a tractable, approximate dynamic program to help determine good orders and withdrawals. This is done by exploring the effects of a measure of the imbalance of inventories at the demand points. Numerical results are included. [ABSTRACT FROM AUTHOR]