[Paper Review] A Sample Path Large Deviation Principle for a Class of Population Processes

William H. Sandholm, Mathias Staudigl|arXiv (Cornell University)|Nov 24, 2015

Game Theory and Applications40 references3 citations

TL;DR

This paper establishes a sample path large deviation principle for Markov chains on discrete grids within the simplex, accounting for boundary effects. By proving joint continuity of the state-dependent Cramér transform, it derives a rigorous rate function for rare event probabilities in population processes, enabling large deviation analysis in systems with state-dependent dynamics and boundaries.

ABSTRACT

We establish a sample path large deviation principle for sequences of Markov chains arising in game theory and other applications. As the state spaces of these Markov chains are discrete grids in the simplex, our analysis must account for the fact that the processes run on a set with a boundary. A key step in the analysis establishes joint continuity properties of the state-dependent Cram´ er transform L( ; ), the running cost appearing in the large deviation principle rate function.

Motivation & Objective

• To develop a large deviation principle for Markov chains modeling population processes in game theory and related applications.
• To address the challenge of boundary effects in discrete state spaces that are grids within the probability simplex.
• To establish joint continuity of the state-dependent Cramér transform, a key component of the rate function in the large deviation principle.
• To provide a rigorous analytical framework for rare event probabilities in finite-state, continuous-time population processes with state-dependent dynamics.

Proposed method

• Analyzes sequences of continuous-time Markov chains evolving on discrete grids in the simplex, representing population distributions.
• Identifies the running cost function in the large deviation rate function as the state-dependent Cramér transform L(·; ·).
• Proves joint continuity of the Cramér transform L(·; ·) in both state and control variables, essential for the large deviation principle.
• Applies sample path large deviation theory to derive a variational representation of the rate function for path probabilities.
• Uses the continuity of the Cramér transform to ensure the rate function is well-defined and lower semicontinuous on the path space.
• Handles boundary effects by carefully analyzing the behavior of the process and the rate function near the simplex boundary.

Experimental results

Research questions

• RQ1 How can a large deviation principle be rigorously established for Markov chains evolving on discrete grids within the simplex, especially near boundaries?
• RQ2 What regularity properties are required for the state-dependent Cramér transform to ensure the validity of the large deviation principle?
• RQ3 How do boundary effects in the state space influence the form and continuity of the rate function in path-space large deviations?
• RQ4 What conditions ensure the joint continuity of the Cramér transform in both state and control variables for population processes?
• RQ5 Can the large deviation principle be applied to model rare events in stochastic population dynamics with state-dependent transition rates?

Key findings

• A sample path large deviation principle is established for Markov chains on discrete grids in the simplex, accounting for boundary effects.
• The joint continuity of the state-dependent Cramér transform L(·; ·) is proven, which is essential for the lower semicontinuity and well-posedness of the rate function.
• The rate function is expressed as a variational formula involving the running cost, with the Cramér transform as its core component.
• The analysis confirms that the large deviation principle holds even when the process approaches the boundary of the simplex.
• The framework applies to population processes in game theory and other applications with state-dependent dynamics.
• The continuity result ensures robustness of the large deviation estimates under small perturbations of the state and control variables.