Abstract

Maslov integration theory allows a process optimization theory to be derived at the same level of generality as stochastic process theory. The approach followed here captures the main idea in forward time, and we therefore introduce Maslov optimization processes such as encountered in maximum likelihood problems. A reversal of time yields optimal control problems of regulation type. We briefly recall the common concepts and, in particular, that the Markov causality principle in this theory is the same as the Bellman optimality principle. We introduce several modes for the convergence of optimization variables and present some classical asymptotic theorems. We derive some optimization martingale properties, such as the inequalities and the (max,-h)-version of the Doob up-crossing lemma which leads to new developments in the field of qualitative studies of optimization processes. Finally we show how some classical large deviation principles translate into this framework and focus on applications to nonlinear filtering.

Introduction

One purpose of this paper is to present a survey of an idempotent measure point of view on optimization theory. The author ([9], [7]) and, independently, Bellalouna ([3]) have introduced the appropriate methodology to derive an optimization theory at the same level of generality as probability and stochastic process theory. This work was clearly inspired by the idempotent measure and integration originally proposed by Maslov [23] as well as the pioneering works of Huillet-Salut [20], Huillet–Rigal–Salut [21] and Del Moral–Huillet–Salut [22].

Sections 1 and 2 are taken from Del Moral [9]. We introduce some basic concepts such as performance spaces, optimization variables, independence and conditioning. Similar developments can be found in Akian–Quadrat–Viot [1], and Del Moral [7], [13] or [14].