Electronic Journal of Probability

Quasi-sure Stochastic Analysis through Aggregation

Mete Soner, Nizar Touzi, and Jianfeng Zhang

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

This work is licensed under aCreative Commons Attribution 3.0 License.


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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