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
Subject: Mathematics
Class: Scientific
Date: Thu, Jul 9, 2009
Location: University of New South Wales, Sydney, Australia
Conference: 1st PRIMA Congress
Subject: Mathematics
Class: Scientific
Abstract:
Let Sn=∑ni=1Xi where {Xi}∞i=1 is a sequence of independent and identically distributed (i.i.d.) of random variables with E[X1]=m. According to the classical law of large number (LLN), the sum Sn/n converges strongly to m. Moreover, the well-known central limit theorem (CLT) tells us that, with m=0 and s2=E[X21], for each bounded and continuous function j we have limnE[j(Sn/√n))]=E[j(X)] with X∼N(0,s2).
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 {Pq:q∈Q}. In this case a robust way to calculate the expectation of a financial loss X is its upper expectation: [\^(E)][X]=supq∈QEq[X] where Eq is the expectation under the probability Pq. The corresponding distribution uncertainty of X is given by Fq(x)=Pq(X≤x), q∈Q. Our main assumptions are:
- The distributions of Xi are within an abstract subset of distributions {Fq(x):q∈Q}, called the distribution uncertainty of Xi, with [′(m)]=[\^(E)][Xi]=supq∫∞−∞xFq(dx) and m=−[\^(E)][−Xi]=infq∫∞−∞xFq(dx).
- Any realization of X1,…,Xn does not change the distributional uncertainty of Xn+1 (a new type of `independence' ).