We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

This paper studies the existence and the order structure of strong Berge equilibrium, a refinement of Nash equilibrium, for games with strategic complementarities à la strong Berge. It is shown that the equilibrium set is a nonempty complete lattice. Moreover, we provide a monotone comparative statics result such that the greatest and the lowest equilibria are increasing.

This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments also has associated increments in the new sense - association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdré, Pérez-Abreu and Surgailis (1998), we show that within the class of multidimensional Gaussian processes, block association of increments is equivalent to supermodularity (in time) of the covariance functions. We also define corresponding versions of weak association, positive association, and negative association. It turns out that the central limit theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling (1986) remains valid, if the weak association is relaxed to the weak association between blocks.

In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-increasing convex order ensures that the conditional expectations are ordered in the convex sense. Finally, the L2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-increasing convex order).

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

This is a study of simple random walks, birth and death processes, and M/M/s queues that have transition probabilities and rates that are sequentially controlled at jump times of the processes. Each control action yields a one-step reward depending on the chosen probabilities or transition rates and the state of the process. The aim is to find control policies that maximize the total discounted or average reward. Conditions are given for these processes to have certain natural monotone optimal policies. Under such a policy for the M/M/s queue, for example, the service and arrival rates are non-decreasing and non-increasing functions, respectively, of the queue length. Properties of these policies and a linear program for computing them are also discussed.