Electronic Journal of Probability

Quasi-sure Stochastic Analysis through Aggregation

Mete Soner, Nizar Touzi, and Jianfeng Zhang

Full-text: Open access

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), 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.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 67, 1844-1879.

Dates
Accepted: 14 October 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820236

Digital Object Identifier
doi:10.1214/EJP.v16-950

Mathematical Reviews number (MathSciNet)
MR2842089

Zentralblatt MATH identifier
1245.60062

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
non-dominated probability measures weak solutions of SDEs uncertain volatility model quasi-sure stochastic analysis

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Soner, Mete; Touzi, Nizar; Zhang, Jianfeng. Quasi-sure Stochastic Analysis through Aggregation. Electron. J. Probab. 16 (2011), paper no. 67, 1844--1879. doi:10.1214/EJP.v16-950. https://projecteuclid.org/euclid.ejp/1464820236


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