Quasi-sure stochastic analysis through aggregation

H. Mete Soner, Nizar Touzi, Jianfeng Zhang

Research output: Contribution to journalArticle

60 Scopus citations

Abstract

This paper is on developing stochastic analysis simultaneously under a general family of probability measures that are not dominated by a single probability measure. The interest in this question originates from the probabilistic representations of fully nonlinear partial differential equations and applications to mathematical finance. The existing literature relies either on the capacity theory (Denis and Martini [5]), or on the underlying nonlinear partial differential equation (Peng [13]). In both approaches, the resulting theory requires certain smoothness, the so called quasi-sure continuity, of the corresponding processes and random variables in terms of the underlying canonical process. In this paper, we investigate this question for a larger class of “non-smooth” processes, but with a restricted family of non-dominated probability measures. For smooth processes, our approach leads to similar results as in previous literature, provided the restricted family satisfies an additional density property.

Original languageEnglish (US)
Pages (from-to)1844-1879
Number of pages36
JournalElectronic Journal of Probability
Volume16
DOIs
StatePublished - Jan 1 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Non-dominated probability measures
  • Quasi-sure stochastic analysis
  • Uncertain volatility model
  • Weak solutions of SDEs

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