### Quasi-sure Stochastic Analysis through Aggregation

**Mete H Soner**

*(Swiss Finance Institute)*

**Nizar Touzi**

*(Ecole Polytechnique Paris)*

**Jianfeng Zhang**

*(University of Southern California)*

#### 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.

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Pages: 1844-1879

Publication Date: October 14, 2011

DOI: 10.1214/EJP.v16-950

#### References

- Barlow, M.T. One-dimensional stochastic differential equation with no strong solution.
*Journal of the London Mathematical Society***26/2**(1982), 335--347. Math. Review 84d:60083 - Carr, P. and Lee, R. Hedging Variance Options on Continuous Semimartingales
*Finance and Stochastics***16/2**(2010), 179--207. Math. Review 2011d:91233 - Cheridito, P., Soner, H.M. and Touzi, N., Victoir, N. Second order BSDE's and fully nonlinear PDE's.
*Communications in Pure and Applied Mathematics*,**60/7**(2007), 1081--1110. Math. Review 2008d:60073 - Dellacherie, C. and Meyer P-A.
*Probabilites et potentiel*, (1980), Chapters V-VIII, Hermann. - Denis, L. and Martini, C. A Theorectical Framework for the Pricing of Contingent Claims in the Presence of Model Uncertainty.
*Annals of Applied Probability***16/2**(2006),827--852. Math. Review 2007j:60100 - Denis, L., Hu, M. and Peng, S. Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths.
*Potential Analysis*,**34/2**(2011), 139--161. Math. Review 2011k:60113 - El Karoui, N. and Quenez, M.-C. Dynamic programming and pricing of contingent claims in an incomplete markets.
*SIAM J. Control. Optim.*,**33/1**, (1995), 29--66. Math. Review 96h:90021 - Fernholz, D. and Karatzas, I. Optimal Arbitrage under Model Uncertainty.
*Annals of Applied Probability*, (2011) to appear. - Fleming, W.H. and Soner, H.M.
*Controlled Markov processes and viscosity solutions*. Applications of Mathematics (New York), 25. Springer-Verlag, New York, (1993). Math. Review 2006e:93002 - Karandikar, R. On pathwise stochastic integration.
*Stochastic Processes and Their Applications.***57**(1995), 11--18. Math. Review 96c:60067 - Karatzas, I. and Shreve, S.
*Brownian Motion and Stochastic Calculus.*2nd Edition, Springer, (1991). Math. Review 92h:60127 - Neveu, J.
*Discrete Parameter Martingales*. North Holland Publishing Company, (1975). Math. Review MR0402915 - Peng, S. G-Brownian motion and dynamic risk measure under volatility uncertainty, arXiv:0711.2834v1, (2007).
- Soner, H. M. and Touzi, N. Dynamic programming for stochastic target problems and geometric flows.
*Journal of European Mathematical Society*,**4/3**(2002), 201--236. Math. Review 2004d:93142 - Soner, H. M. Touzi, N. and Zhang, J. Martingale representation theorem for $G-$expectation.
*Stochastic Processes and their Applications*,**121**(2011), 265--287. Math. Review MR2746175 - Soner, H. M. Touzi, N. and Zhang, J. Dual Formulation of Second Order Target Problems. (2009) arXiv:1003.6050.
- Soner, H. M. Touzi, N. and Zhang, J. Wellposedness of second order backward SDEs.
*Probability Theory and Related Fields*, (2011), to appear. - Stroock, D.W. and Varadhan, S.R.S.
*Multidimensional Diffusion Processes*. (1979), Springer-Verlag, Berlin, Heidelberg, New York. Math. Review 81f:60108

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