Law of Large Number and Central Limit Theorem under Uncertainty, the related New Itô's Calculus and Applications to Risk Measures

Speaker: 
Shige Peng
Date: 
Thu, Jul 9, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 

Let TeX Embedding failed! where TeX Embedding failed! is a sequence of independent and identically distributed (i.i.d.) of random variables with TeX Embedding failed!. According to the classical law of large number (LLN), the sum TeX Embedding failed! converges strongly to TeX Embedding failed!. Moreover, the well-known central limit theorem (CLT) tells us that, with TeX Embedding failed! and TeX Embedding failed!, for each bounded and continuous function TeX Embedding failed! we have TeX Embedding failed! with TeX Embedding failed!.

These two fundamentally important results are widely used in probability, statistics, data analysis as well as in many practical situation such as financial pricing and risk controls. They provide a strong argument to explain why in practice normal distributions are so widely used. But a serious problem is that the i.i.d. condition is very difficult to be satisfied in practice for the most real-time processes for which the classical trials and samplings becomes impossible and the uncertainty of probabilities and/or distributions cannot be neglected.

In this talk we present a systematical generalization of the above LLN and CLT. Instead of fixing a probability measure P, we only assume that there exists a uncertain subset of probability measures TeX Embedding failed!. In this case a robust way to calculate the expectation of a financial loss TeX Embedding failed! is its upper expectation: TeX Embedding failed! where TeX Embedding failed! is the expectation under the probability TeX Embedding failed!. The corresponding distribution uncertainty of TeX Embedding failed! is given by TeX Embedding failed!, TeX Embedding failed!. Our main assumptions are:

  1. The distributions of TeX Embedding failed! are within an abstract subset of distributions TeX Embedding failed!, called the distribution uncertainty of TeX Embedding failed!, with TeX Embedding failed! and TeX Embedding failed!.
  2. Any realization of TeX Embedding failed! does not change the distributional uncertainty of TeX Embedding failed! (a new type of `independence' ).

Our new LLN is: for each linear growth continuous function TeX Embedding failed! we have

TeX Embedding failed!

Namely, the distribution uncertainty of TeX Embedding failed! is, approximately, TeX Embedding failed!.

In particular, if TeX Embedding failed!, then TeX Embedding failed! converges strongly to 0. In this case, if we assume furthermore that TeX Embedding failed! and TeX Embedding failed!, TeX Embedding failed!. Then we have the following generalization of the CLT:

TeX Embedding failed!

Here TeX Embedding failed! stands for a distribution uncertainty subset and TeX Embedding failed! its the corresponding upper expectation. The number TeX Embedding failed! can be calculated by defining TeX Embedding failed! which solves the following PDE TeX Embedding failed!, with TeX Embedding failed!

An interesting situation is when TeX Embedding failed! is a convex function, TeX Embedding failed! with TeX Embedding failed!. But if TeX Embedding failed! is a concave function, then the above TeX Embedding failed! has to be replaced by TeX Embedding failed!. This coincidence can be used to explain a well-known puzzle: many practitioners, particularly in finance, use normal distributions with `dirty' data, and often with successes. In fact, this is also a high risky operation if the reasoning is not fully understood. If TeX Embedding failed!, then TeX Embedding failed! which is a classical normal distribution. The method of the proof is very different from the classical one and a very deep regularity estimate of fully nonlinear PDE plays a crucial role.

A type of combination of LLN and CLT which converges in law to a more general TeX Embedding failed!-distributions have been obtained. We also present our systematical research on the continuous-time counterpart of the above `G-normal distribution', called G-Brownian motion and the corresponding stochastic calculus of Itô's type as well as its applications.