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

Date: Thu, Jul 9, 2009

Location: University of New South Wales, Sydney, Australia

Conference: 1st PRIMA Congress

Subject: Mathematics

Class: Scientific

### Abstract:

Let $S_n= \sum_{i=1}^n X_i$ where $\{X_i\}_{i=1}^\infty$ is a sequence of independent and identically distributed (i.i.d.) of random variables with $E[X_1]=m$. According to the classical law of large number (LLN), the sum $S_n/n$ converges strongly to $m$. Moreover, the well-known central limit theorem (CLT) tells us that, with $m = 0$ and $s^2=E[X_1^2]$, for each bounded and continuous function $j$ we have $\lim_n E[j(S_n/\sqrt{n}))]=E[j(X)]$ with $X \sim N(0, s^2)$.
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 $\{P_q:q \in Q\}$. In this case a robust way to calculate the expectation of a financial loss $X$ is its upper expectation: $[\^\,(\mathbf{E})][X]=\sup_{q \in Q} E_q[X]$ where $E_q$ is the expectation under the probability $P_q$. The corresponding distribution uncertainty of $X$ is given by $F_q(x)=P_q(X \leq x)$, $q \in Q$. Our main assumptions are:

- The distributions of $X_i$ are within an abstract subset of distributions $\{F_q(x):q \in Q\}$, called the distribution uncertainty of $X_i$, with $['(m)]=[\^(\mathbf{E})][X_i]=\sup_q\int_{-\infty}^\infty xF_q(dx)$ and $m=-[\^\,(\mathbf{E})][-X_i]=\inf_q \int_{-\infty}^\infty x F_q(dx)$.
- Any realization of $X_1, \ldots, X_n$ does not change the distributional uncertainty of $X_{n+1}$ (a new type of `independence' ).