2022/05/19 by Amarjit Budhiraja, Budhiraja, Amarjit, Adam Waterbury +1 · 1 citation
Mathematics · Economics, Econometrics and Finance · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and financial applications #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2205.09291
Let A be a transition probability kernel on a finite state space Δo =\1, … , d\ such that A(x,y)>0 for all x,y ∈ Δo. Consider a reinforced chain given as a sequence \Xn, n ∈ ℕ0\ of Δo-valued random variables, defined recursively according to, Ln = (1)/(n)∑i=0n-1 δXi, P(Xn+1 ∈ ⋅ | X0, …, Xn) = Ln A(⋅). We establish a large deviation principle for \Ln\. The rate function takes a strikingly different form than the Donsker-Varadhan rate function associated with the empirical measure of the Markov chain with transition kernel A and is described in terms of a novel deterministic infinite horizon discounted cost control problem with an associated linear controlled dynamics and a nonlinear running cost involving the relative entropy function. Proofs are based on an analysis of time-reversal of controlled dynamics in representations for log-transforms of exponential moments, and on weak convergence methods.