• Corpus ID: 15546914

# A New Central Limit Theorem under Sublinear Expectations

@article{Peng2008ANC,
title={A New Central Limit Theorem under Sublinear Expectations},
author={Shige Peng},
journal={arXiv: Probability},
year={2008}
}
• S. Peng
• Published 18 March 2008
• Mathematics
• arXiv: Probability
We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty.
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## References

SHOWING 1-10 OF 25 REFERENCES

### Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths

• Mathematics
• 2008
In this paper we give some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation—G-expectation. Many results can

### G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty

We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Ito's type (Ito's integral, Ito's formula, Ito's equation) through the

### NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS

This paper deals with nonlinear expectations. The author obtains a nonlinear generalization of the well-known Kolmogorov's consistent theorem and then use it to construct filtration-consistent

### NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS

This paper deals with nonlinear expectations. The author obtains a nonlinear generalization of the well-known Kolmogorov's consistent theorem and then use it to construct filtration-consistent

### G -Expectation, G -Brownian Motion and Related Stochastic Calculus of Itô Type

We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution.

### A strong law of large numbers for capacities

• Mathematics
• 2005
We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the

### A THEORETICAL FRAMEWORK FOR THE PRICING OF CONTINGENT CLAIMS IN THE PRESENCE OF MODEL UNCERTAINTY

• Mathematics
• 2006
The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a

### User’s guide to viscosity solutions of second order partial differential equations

• Mathematics
• 1992
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence

### Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims

We will study the following problem. Let Xt, t ∈ [0, T], be an Rd–valued process defined on a time interval t ∈ [0, T]. Let Y be a random value depending on the trajectory of X. Assume that, at each