This paper presents a model of capital accumulation for a large number of heterogenous producer-consumers in an exchange space in which interactions depend on agents' positions. Each agent is described by his production, consumption, stock of capital, as well as the position he occupies in this abstract space. Each agent produces one differentiated good whose price is fixed by market clearing conditions. Production functions are Cobb-Douglas, and capital stocks follow the standard capital accumulation dynamic equation. Agents consume all goods but have a preference for goods produced by their closest neighbors. Agents in the exchange space are subject both to attractive and repulsive forces. Exchanges drive agents closer, but beyond a certain level of proximity, agents will tend to crowd out more distant agents. The present model uses a formalism based on statistical field theory developed earlier by the authors. This approach allows the analytical treatment of economic models with an arbitrary number of agents, while preserving the system's interactions and complexity at the individual level. Our results show that the dynamics of capital accumulation and agents' position in the exchange space are correlated. Interactions in the exchange space induce several phases of the system. A first phase appears when attractive forces are limited. In this phase, an initial central position in the exchange space favors capital accumulation in average and leads to a higher level of capital, while agents far from the center will experience a slower accumulation process. A high level of initial capital drives agents towards a central position, i.e. improve the terms of their exchanges: they experience a higher demand and higher prices for their product. As usual, high capital productivity favors capital accumulation, while higher rates of capital depreciation reduce capital stock. In a second phase, attractive forces are predominant. The previous results remain, but an additional threshold effect appears. Even though no restriction was imposed initially on the system, two types of agents emerge, depending on their initial stock of capital. One type of agents will remain above the capital threshold and occupy and benefit from a central position. The other type will remain below the threshold, will not be able to break it and will remain at the periphery of the exchange space. In this phase, capital distribution is less homogenous than in the first phase.

Companies try to maximize their profits by recovering returned products of highly uncertain quality and quantity. In this paper, a reverse logistics network for an Original Equipment Manufacturer (OEM) is presented. Returned products are selected for remanufacturing or scrapping, based on their quality and proportional prices are offered to customers. A Mixed Integer Non-linear Programming (MINLP) model is proposed to determine the location of collection centers, the optimum price of returned products and the sorting policy. The risk in the objective function is measured using the Conditional Value at Risk (CVaR) metric. CVaR measures the risk of an investment in a conservative way by considering the maximum lost. The results are analyzed for various values of the risk parameters ({\alpha}, and {\lambda}). These parameters indicate that considering risk affects prices, the classification of returned products, the location of collection centers and, consequently, the objective function. The model performs more conservatively when the weight of the CVaR part ({\lambda}) and the value of the confidence level {\alpha} are increased. The results show that better profits are obtained when we take CVaR into account.

We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price functions between adjacent timesteps. We introduce the least squares residual of the associated backward stochastic differential equation as the loss function. Our proposed framework yields prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer. Our numerical simulations demonstrate these contributions, and show that the proposed neural network framework outperforms state-of-the-art approaches in high dimensions.

Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures, and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the "times of fragility" of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells "synchronize") explicitly in terms of the dynamics of the driving processes, the current distribution of the particles' values, and the topology of the underlying network (represented by its Perron-Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes: i.e., the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder's fixed-point theorem for the Skorokhod space with the M1 topology, and the application of the max-plus algebra to the equilibrium version of the network flow problem.