We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme th
at produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework.
Adopting a probabilistic approach we determine the optimal dividend payout policy of a firm whose surplus process follows a controlled arithmetic Brownian motion and whose cash-flows are discounted at a stochastic dynamic rate. Dividends can be paid
to shareholders at unrestricted rates so that the problem is cast as one of singular stochastic control. The stochastic interest rate is modelled by a Cox-Ingersoll-Ross (CIR) process and the firms objective is to maximize the total expected flow of discounted dividends until a possible insolvency time. We find an optimal dividend payout policy which is such that the surplus process is kept below an endogenously determined stochastic threshold expressed as a decreasing continuous function $r mapsto b(r)$ of the current interest rate value. We also prove that the value function of the singular control problem solves a variational inequality associated to a second-order, non-degenerate elliptic operator, with a gradient constraint.
In this paper we present a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a rate
of convergence in a suitable $L^1$-norm and we implement the scheme numerically. To the best of our knowledge this is the first paper to study (and implement) numerical solutions of SDEs whose drift lives in a space of distributions. As a byproduct we also obtain an estimate of the convergence rate for a numerical scheme applied to SDEs with drift in $L^p$-spaces with $pin(1,infty)$.
Recent experimental results demonstrated the generation of a quantum superpositon (MQS), involving a number of photons in excess of 5x10^4, which showed a high resilience to losses. In order to perform a complete analysis on the effects of de-coheren
ce on this multiphoton fields, obtained through the Quantum Injected Optical Parametric Amplifier (QIOPA), we invesigate theoretically the evolution of the Wigner functions associated to these states in lossy conditions. Recognizing the presence of negative regions in the W-representation as an evidence of non-classicality, we focus our analysis on this feature. A close comparison with the MQS based on coherent states allows to identify differences and analogies.
We investigate the multiphoton states generated by high-gain optical parametric amplification of a single injected photon, polarization encoded as a qubit. The experiment configuration exploits the optimal phase-covariant cloning in the high gain reg
ime. The interference fringe pattern showing the non local transfer of coherence between the injected qubit and the mesoscopic amplified output field involving up to 4000 photons has been investigated. A probabilistic new method to extract full information about the multiparticle output wavefunction has been implemented.
In 1981 N. Herbert proposed a gedanken experiment in order to achieve by the First Laser Amplified Superluminal Hookup (FLASH) a faster than light communication (FTL) by quantum nonlocality. The present work reports the first experimental realization
of that proposal by the optical parametric amplification of a single photon belonging to an entangled EPR pair into an output field involving 5 x 10^3 photons. A thorough theoretical and experimental analysis explains in general and conclusive terms the precise reasons for the failure of the FLASH program as well as of any similar FTL proposals.
