Local asymptotics for controlled martingales

Scott N. Armstrong, Ofer Zeitouni

Research output: Contribution to journalArticle

Abstract

We consider controlled martingales with bounded steps where the controller is allowed at each step to choose the distribution of the next step, and where the goal is to hit a fixed ball at the origin at time n. We show that the algebraic rate of decay (as n increases to infinity) of the value function in the discrete setup coincides with its continuous counterpart, provided a reachability assumption is satisfied. We also study in some detail the uniformly elliptic case and obtain explicit bounds on the rate of decay. This generalizes and improves upon several recent studies of the one dimensional case, and is a discrete analogue of a stochastic control problem recently investigated in Armstrong and Trokhimtchouck [Calc. Var. Partial Differential Equations 38 (2010) 521-540].

Original languageEnglish (US)
Pages (from-to)1467-1494
Number of pages28
JournalAnnals of Applied Probability
Volume26
Issue number3
DOIs
StatePublished - Jun 2016

Keywords

  • Martingale
  • Nonlinear parabolic equation
  • Stochastic control

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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