This paper is the companion article to [Ann. Probab. 39 (2011) 779–856]. We consider a discrete elliptic equation on the d-dimensional lattice ℤ^d with random coefficients A of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric “homogenized” matrix A_hom = a_homId is characterized by ξ ⋅ A_homξ = 〈(ξ + ∇ϕ) ⋅ A(ξ + ∇ϕ)〉 for any direction ξ ∈ ℝ^d, where the random field ϕ (the “corrector”) is the unique solution of −∇* ⋅ A(ξ + ∇ϕ) = 0 in ℤ^d such that ϕ(0) = 0, ∇ϕ is stationary and 〈∇ϕ〉 = 0, 〈⋅〉 denoting the ensemble average (or expectation).

In order to approximate the homogenized coefficients A_hom, the corrector problem is usually solved in a box Q_L = [−L, L)^d of size 2L with periodic boundary conditions, and the space averaged energy on Q_L defines an approximation A_L of A_hom. Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation A_L converges almost surely to A_hom as L ↑ ∞. In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size 2L, but replace the elliptic operator by T⁻¹ − ∇^∗ ⋅ A∇ with (typically) T ∼ L², as standard in the homogenization literature. We then replace the ensemble average by a space average on Q_L, and estimate the overall error on the homogenized coefficients in terms of L and T.

In the recent past, there has been a concerted effort to develop mathematical models for real-world networks and to analyze various dynamics on these models. One particular problem of significant importance is to understand the effect of random edge lengths or costs on the geometry and flow transporting properties of the network. Two different regimes are of great interest, the weak disorder regime where optimality of a path is determined by the sum of edge weights on the path and the strong disorder regime where optimality of a path is determined by the maximal edge weight on the path. In the context of the stochastic mean-field model of distance, we provide the first mathematically tractable model of weak disorder and show that no transition occurs at finite temperature. Indeed, we show that for every finite temperature, the number of edges on the minimal weight path (i.e., the hopcount) is Θ(log n) and satisfies a central limit theorem with asymptotic means and variances of order Θ(log n), with limiting constants expressible in terms of the Malthusian rate of growth and the mean of the stable-age distribution of an associated continuous-time branching process. More precisely, we take independent and identically distributed edge weights with distribution E^s for some parameter s > 0, where E is an exponential random variable with mean 1. Then the asymptotic mean and variance of the central limit theorem for the hopcount are s log n and s² log n, respectively. We also find limiting distributional asymptotics for the value of the minimal weight path in terms of extreme value distributions and martingale limits of branching processes.

We consider a queueing network in which there are constraints on which queues may be served simultaneously; such networks may be used to model input-queued switches and wireless networks. The scheduling policy for such a network specifies which queues to serve at any point in time. We consider a family of scheduling policies, related to the maximum-weight policy of Tassiulas and Ephremides [IEEE Trans. Automat. Control 37 (1992) 1936–1948], for single-hop and multihop networks. We specify a fluid model and show that fluid-scaled performance processes can be approximated by fluid model solutions. We study the behavior of fluid model solutions under critical load, and characterize invariant states as those states which solve a certain network-wide optimization problem. We use fluid model results to prove multiplicative state space collapse. A notable feature of our results is that they do not assume complete resource pooling.

There has recently been considerable interest in design of low-complexity, myopic, distributed and stable scheduling algorithms for constrained queueing network models that arise in the context of emerging communication networks. Here we consider two representative models. One, a queueing network model that captures randomly varying number of packets in the queues present at a collection of wireless nodes communicating through a shared medium. Two, a buffered circuit switched network model for an optical core of future internet to capture the randomness in calls or flows present in the network. The maximum weight scheduling algorithm proposed by Tassiulas and Ephremides [IEEE Trans. Automat. Control 37 (1992) 1936–1948], leads to a myopic and stable algorithm for the packet-level wireless network model. But computationally it is expensive (NP-hard) and centralized. It is not applicable to the buffered circuit switched network due to the requirement of nonpreemption of the calls in the service. As the main contribution of this paper, we present a stable scheduling algorithm for both of these models. The algorithm is myopic, distributed and performs few logical operations at each node per unit time.

For a stochastic factor model we maximize the long-term growth rate of robust expected power utility with parameter λ ∈ (0, 1). Using duality methods the problem is reformulated as an infinite time horizon, risk-sensitive control problem. Our results characterize the optimal growth rate, an optimal long-term trading strategy and an asymptotic worst-case model in terms of an ergodic Bellman equation. With these results we propose a duality approach to a “robust large deviations” criterion for optimal long-term investment.

The framework of this paper is that of risk measuring under uncertainty which is when no reference probability measure is given. To every regular convex risk measure on $\mathcal{C}_{b}(\Omega)$, we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless nonpositive elements of $\mathcal{C}_{b}(\Omega)$. We prove that the convex risk measure has a dual representation with a countable set of probability measures absolutely continuous with respect to a certain probability measure in this class. To get these results we study the topological properties of the dual of the Banach space L¹(c) associated to a capacity c.

As application, we obtain that every G-expectation $\mathbb{E}$ has a representation with a countable set of probability measures absolutely continuous with respect to a probability measure P such that P(|f|) = 0 if and only iff $\mathbb{E}(|f|)=0$. We also apply our results to the case of uncertain volatility.

This paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model for an investor with power utility and a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk.

In models with several assets and a single state variable, long-run portfolios and risk premia admit explicit formulas up the solution of an ordinary differential equation which characterizes the principal eigenvalue of an elliptic operator. Multiple state variables lead to a quasilinear partial differential equation which is solvable for many models of interest.

The paper derives the long-run optimal portfolio and the long-run optimal pricing measures depending on relative risk aversion, as well as their finite-horizon performance.

In this paper we consider a class of BSDEs with drivers of quadratic growth, on a stochastic basis generated by continuous local martingales. We first derive the Markov property of a forward–backward system (FBSDE) if the generating martingale is a strong Markov process. Then we establish the differentiability of a FBSDE with respect to the initial value of its forward component. This enables us to obtain the main result of this article, namely a representation formula for the control component of its solution. The latter is relevant in the context of securitization of random liabilities arising from exogenous risk, which are optimally hedged by investment in a given financial market with respect to exponential preferences. In a purely stochastic formulation, the control process of the backward component of the FBSDE steers the system into the random liability and describes its optimal derivative hedge by investment in the capital market, the dynamics of which is given by the forward component. The representation formula of the main result describes this delta hedge in terms of the derivative of the BSDEs solution process on the one hand and the correlation structure of the internal uncertainty captured by the forward process and the external uncertainty responsible for the market incompleteness on the other hand. The formula extends the scope of validity of the results obtained by several authors in the Brownian setting. It is designed to extend a genuinely stochastic representation of the optimal replication in cross hedging insurance derivatives from the classical Black–Scholes model to incomplete markets on general stochastic bases. In this setting, Malliavin’s calculus which is required in the Brownian framework, is replaced by new tools based on techniques related to a calculus of quadratic covariations of basis martingales.

We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to ∞. The random fields considered include costs of assignments, weights of Hamiltonian cycles and spanning trees, energies of directed polymers, locations of particles in the branching random walk, as well as energies in the Sherrington–Kirkpatrick and Edwards–Anderson models. The distribution of levels in all models listed above is shown to be essentially the same as in a stationary Gaussian process with regularly varying nonsummable covariance function. This type of behavior is different from the Brownian bridge-type limit known for independent or stationary weakly dependent sequences of random variables.

We study utility maximization for power utility random fields with and without intermediate consumption in a general semimartingale model with closed portfolio constraints. We show that any optimal strategy leads to a solution of the corresponding Bellman equation. The optimal strategies are described pointwise in terms of the opportunity process, which is characterized as the minimal solution of the Bellman equation. We also give verification theorems for this equation.

We consider optimal control of a multi-class queue in the Halfin–Whitt regime, and revisit the notion of asymptotic optimality and the associated optimality gaps. The existing results in the literature for such systems provide asymptotically optimal controls with optimality gaps of $o(\sqrt{n})$ where n is the system size, for example, the number of servers. We construct a sequence of asymptotically optimal controls where the optimality gap grows logarithmically with the system size. Our analysis relies on a sequence of Brownian control problems, whose refined structure helps us achieve the improved optimality gaps.