topics in statistical science. The corresponding chapters of this volume range from the foundations of parametric, semi-parametric, and nonparametric inference through current work on stochastic modeling in financial econometrics, epidemics, and eco-hydrology to the prognosis for the treatment of breast cancer, with excursions on biostatistics, social statistics, and statistical computing.” The invited speakers are all prominent researchers as well as friends, colleagues, collaborators, and former students of David Cox. The articles in this book demonstrate richness of statistical science in its breadth, depth, applicability, quality, and its impact on real life. These articles collectively reflect David Cox’s strong influence on modern statistics. The editors acknowledged in the Preface, “We are most grateful to the authors of these chapters, which testify not only to the extraordinary breadth of David’s interests but also to his characteristic desire to look forward rather than backward.” The book has 12 chapters with references from all chapters presented together at the end in the Bibliography. This moderately priced collection of articles is a valuable resource for all research statisticians, users of statistics for research and practice in a wide spectrum of disciplines, and statistics graduate students.

Measures, Integrals and Martingales

In this paper, we consider discrete-time dynamic games of the mean-field type with a finite number, N, of agents subject to an infinite-horizon discounted-cost optimality criterion. The state space of each agent is a locally compact Polish space. At each time, the agents are coupled through the empirical distribution of their states, which affects both the agents' individual costs and their state transition probabilities. We introduce the solution concept of Markov-Nash equilibrium, under which a policy is player-by-player optimal in the class of all Markov policies. Under mild assumptions, we demonstrate the existence of a mean-field equilibrium in the infinite-population limit, N → 1, and then show that the policy obtained from the mean-field equilibrium is approximately Markov-Nash when the number of agents N is sufficiently large.

Markov-Nash equilibria in mean-field games with discounted cost

Establishing the existence of Nash equilibria for partially observed stochastic dynamic games is known to be quite challenging, with the difficulties stemming from the noisy nature of the measureme...

Approximate nash equilibria in partially observed stochastic games with mean-field interactions

In this paper, we study a class of discrete-time mean-field games under the infinite-horizon risk-sensitive optimality criterion. Risk sensitivity is introduced for each agent (player) via an expon...

Approximate Markov-Nash Equilibria for Discrete-Time Risk-Sensitive Mean-Field Games

We consider in this paper a general class of discrete-time partially-observed mean-field games with Polish state, action, and measurement spaces and with risk-sensitive (exponential) cost functions which capture the risk-averse behaviour of each agent. As standard in mean-field game models, here each agent is weakly coupled with the rest of the population through its individual cost and state dynamics via the empirical distribution of the states. We first establish the mean-field equilibrium in the infinite-population limit by first transforming the risk-sensitive problem to one with risk-neutral (that is, additive instead of multiplicative) cost function, and then employing the technique of converting the underlying original partially-observed stochastic control problem to a fully observed one on the belief space and the principle of dynamic programming. Then, we show that the mean-field equilibrium policy, when adopted by each agent, forms an approximate Nash equilibrium for games with sufficiently many agents.

Partially-Observed Discrete-Time Risk-Sensitive Mean-Field Games

We study long time behavior of a discrete time weakly interacting particle system, and the corresponding nonlinear Markov process in , described in terms of a general stochastic evolution equation. In a setting where the state space of the particles is compact such questions have been studied in previous works, however, for the case of an unbounded state space very few results are available. Under suitable assumptions on the problem data we study several time asymptotic properties of the N-particle system and the associated nonlinear Markov chain. In particular, we show that the evolution equation for the law of the nonlinear Markov chain has a unique fixed point and starting from an arbitrary initial condition convergence to the fixed point occurs at an exponential rate. The empirical measure μNn of the N-particles at time n is shown to converge to the law μn of the nonlinear Markov process at time n, in the Wasserstein-1 distance, in L1, as N → ∞, uniformly in n. Several consequences of this uniform con...

Long Time Results for a Weakly Interacting Particle System in Discrete Time

Regime switching processes have proved to be indispensable in the modeling of various phenomena, allowing model parameters that traditionally were considered to be constant to fluctuate in a Markovian manner in line with empirical findings. We study diffusion processes of Ornstein–Uhlenbeck type where the drift and diffusion coefficients $a$ and $b$ are functions of a Markov process with a stationary distribution $\pi $ on a countable state space. Exact long time behavior is determined for the three regimes corresponding to the expected drift: $E_{\pi }a(\cdot )>0$, $=0,<0$, respectively. Alongside we provide exact time limit results for integrals of form $\int _{0}^{t}b^{2}(X_{s})e^{-2\int _{s}^{t}a(X_{r})\,dr}\,ds$ for the three different regimes. Finally, we demonstrate natural applications of the findings in terms of Cox–Ingersoll–Ross diffusion and deterministic SIS epidemic models in Markovian environments. The time asymptotic behaviors are naturally expressed in terms of solutions to the well-studied fixed-point equation in law $X\stackrel{d}{=}AX+B$ with $X\perp \!\!\!\!\perp (A,B)$.

/pdf/exact-long-time-behavior-of-some-regime-switching-stochastic-358wr9heyo.pdf

Exact long time behavior of some regime switching stochastic processes

Regime switching processes have proved to be indispensable in the modeling of various phenomena, allowing model parameters that traditionally were considered to be constant to fluctuate in a Markovian manner in line with empirical findings. We study diffusion processes of Ornstein-Uhlenbeck type where the drift and diffusion coefficients $a$ and $b$ are functions of a Markov process with a stationary distribution $\pi$ on a countable state space. Exact long time behavior is determined for the three regimes corresponding to the expected drift: $E_{\pi}a(\cdot)>0,=0,<0$, respectively. Alongside we provide exact time limit results for integrals of form $\int_{0}^{t}b^{2}(X_{s})e^{-2\int_{s}^{t}a(X_{r})dr}ds$ for the three different regimes. Finally, we demonstrate natural applications of the findings in terms of Cox-Ingersoll-Ross diffusion and deterministic SIS epidemic models in Markovian environments. Exact long time behaviors are naturally expressed in terms of solutions to the well-studied fixed-point equation in law $X\stackrel{d}{=}AX+B$ with $X \indep (A,B)$.

Exact long time behavior of some regime switching stochastic processes.

We study long time behavior of a discrete time weakly interacting particle system, and the corresponding nonlinear Markov process in $\mathbb{R}^d$, described in terms of a general stochastic evolution equation. In a setting where the state space of the particles is compact such questions have been studied in previous works, however for the case of an unbounded state space very few results are available. Under suitable assumptions on the problem data we study several time asymptotic properties of the $N$-particle system and the associated nonlinear Markov chain. In particular we show that the evolution equation for the law of the nonlinear Markov chain has a unique fixed point and starting from an arbitrary initial condition convergence to the fixed point occurs at an exponential rate. The empirical measure $\mu_{n}^{N}$ of the $N$-particles at time $n$ is shown to converge to the law $\mu_{n}$ of the nonlinear Markov process at time $n$, in the Wasserstein-1 distance, in $L^{1}$, as $N\to \infty$, uniformly in $n$. Several consequences of this uniform convergence are studied, including the interchangeability of the limits $n\to \infty$ and $N\to\infty$ and the propagation of chaos property at $n = \infty$. Rate of convergence of $\mu_{n}^{N}$ to $\mu_{n}$ is studied by establishing uniform in time polynomial and exponential probability concentration estimates.

A system of $N$ particles in a chemical medium in $\mathbb{R}^{d}$ is studied in a discrete time setting. Underlying interacting particle system in continuous time can be expressed as \begin{eqnarray} dX_{i}(t) &=&[-(I-A)X_{i}(t) + \bigtriangledown h(t,X_{i}(t))]dt + dW_{i}(t), \,\, X_{i}(0)=x_{i}\in \mathbb{R}^{d}\,\,\forall i=1,\ldots,N\nonumber\\ \frac{\partial}{\partial t} h(t,x)&=&-\alpha h(t,x) + D\bigtriangleup h(t,x) +\frac{\beta}{n} \sum_{i=1}^{N} g(X_{i}(t),x),\quad h(0,\cdot) = h(\cdot).\label{main} \end{eqnarray} where $X_{i}(t)$ is the location of the $i$th particle at time $t$ and $h(t,x)$ is the function measuring the concentration of the medium at location $x$ with $h(0,x) = h(x)$. In this article we describe a general discrete time non-linear formulation of the aforementioned model and a strongly coupled particle system approximating it. Similar models have been studied before (Budhiraja et al.(2011)) under a restrictive compactness assumption on the domain of particles. In current work the particles take values in $\R^{d}$ and consequently the stability analysis is particularly challenging. We provide sufficient conditions for the existence of a unique fixed point for the dynamical system governing the large $N$ asymptotics of the particle empirical measure. We also provide uniform in time convergence rates for the particle empirical measure to the corresponding limit measure under suitable conditions on the model.

Abhishek Pal Majumder

Papers

Long Time Results for a Weakly Interacting Particle System in Discrete Time

Exact long time behavior of some regime switching stochastic processes

Exact long time behavior of some regime switching stochastic processes.

Long Time Results for a Weakly Interacting Particle System in Discrete Time

Quantitative evaluation of an active Chemotaxis model in Discrete time