Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures

We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable con dence intervals on the optimal value of such stochastic programs using the Robust Stochastic Approximation and the Stochastic Mirror Descent (SMD) algorithms. When the objective functions are uniformly convex, we also propose a multistep extension of the Stochastic Mirror Descent algorithm and obtain con dence intervals on both the optimal values and optimal solutions. Numerical simulations show that our con dence intervals are much less conservative and are quicker to compute than previously obtained con dence intervals for SMD and that the multistep Stochastic Mirror Descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart. Our con dence intervals are also more reliable than asymptotic con dence intervals when the sample size is not much larger than the problem size.

Convexity in multidimensional mechanism design

The present article initiates a systematic study of the behavior of a strictly increasing, C2 , utility function u(a), seen as a function of agents' types, a, when the set of types, A, is a compact, convex subset of iRm . When A is a m-dimensional rectangle it shows that there is a diffeomorphism of A such that the function U = u o H is strictly increasing, C2 , and strictly convexo Moreover, when A is a strictly convex leveI set of a nowhere singular function, there exists a change of coordinates H such that B = H-1(A) is a strictly convex set and U = u o H : B ~ iR is a strictly convex function, as long as a characteristic number of u is smaller than a characteristic number of A. Therefore, a utility function can be assumed convex in agents' types without loss of generality in a wide variety of economic environments.

On the differentiability of the consumer demand function

For strictly quasi concave differentiable utility functions, demand is shown to be differentiable almost everywhere if marginal utilities are pointwise Lipschitzian. For concave utility functions, demand is differentiable almost everywhere in the case of differentiable additively separable utility or in the case of quasi-linear utility.

Integral representation with convex capacities that are squeeze of (additive) probability measures

In this paper I will investigate the conditions under which a convex capacity (or a non-additive probability which exhibts uncertainty aversion) can be represented as a squeeze of a(n) (additive) probability measure associate to an uncertainty aversion function. Then I will present two alternatives forrnulations of the Choquet integral (and I will extend these forrnulations to the Choquet expected utility) in a parametric approach that will enable me to do comparative static exercises over the uncertainty aversion function in an easy way.

A critical value function approach, with an application to persistent time-series

Researchers often rely on the t-statistic to make inference on parameters in statistical models. It is common practice to obtain critical values by simulation techniques. This paper proposes a novel numerical method to obtain an approximately similar test. This test rejects the null hypothesis when the test statistic islarger than a critical value function (CVF) of the data. We illustrate this procedure when regressors are highly persistent, a case in which commonly-used simulation methods encounter dificulties controlling size uniformly. Our approach works satisfactorily, controls size, and yields a test which outperforms the two other known similar tests.