Quasi-sure Stochastic Analysis through Aggregation

Mete H Soner (Swiss Finance Institute)
Nizar Touzi (Ecole Polytechnique Paris)
Jianfeng Zhang (University of Southern California)


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), or on the underlying nonlinear partial differential equation (Peng). 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.

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Pages: 1844-1879

Publication Date: October 14, 2011

DOI: 10.1214/EJP.v16-950


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