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

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

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 . In this case a robust way to calculate the expectation of a financial loss is its upper expectation: where is the expectation under the probability . The corresponding distribution uncertainty of is given by , . Our main assumptions are:

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

Our new LLN is: for each linear growth continuous function we have

Namely, the distribution uncertainty of is, approximately, .

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

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

An interesting situation is when is a convex function, with . But if is a concave function, then the above has to be replaced by . 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 , then 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 -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.