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

Shige Peng
Thu, Jul 9, 2009
University of New South Wales, Sydney, Australia
1st PRIMA Congress

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:

  1. 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) $.
  2. Any realization of $ X_1, \ldots, X_n $ does not change the distributional uncertainty of $ X_{n+1} $ (a new type of `independence' ).

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

$$\lim_{n\to\infty} \^{\mathbf{E}}[j(S_n/n)] = \sup_{m\leq v\leq ['(m)]} j(v)$$

Namely, the distribution uncertainty of $ S_n/n $ is, approximately, $ \{ d_v:m \leq v \leq ['(m)]\} $.

In particular, if $ m=['(m)]=0 $, then $ S_n/n $ converges strongly to 0. In this case, if we assume furthermore that $ ['(s)]2=[\^\,(\mathbf{E})][X_i^2] $ and $ s_2=-[\^\,(\mathbf{E})][-X_i^2] $, $ i=1, 2, \ldots $. Then we have the following generalization of the CLT:

$$\lim_{n\to\infty} [j(Sn/\sqrt{n})]= \^{\mathbf{E}}[j(X)], L(X)\in N(0,[s^2,\overline{s}^2]).$$

Here $ N(0, [s^2, ['(s)]^2]) $ stands for a distribution uncertainty subset and $ [\^(E)][j(X)] $ its the corresponding upper expectation. The number $ [\^(E)][j(X)] $ can be calculated by defining $ u(t, x):=[^(\mathbf{E})][j(x+\sqrt{tX})] $ which solves the following PDE $ \partial_t u= G(u_{xx}) $, with $ G(a):=[1/2](['(s)]^2a^+-s^2a^-). $

An interesting situation is when $ j $ is a convex function, $ [\^\,(\mathbf{E})][j(X)]=E[j(X_0)] $ with $ X_0 \sim N(0, ['(s)]^2) $. But if $ j $ is a concave function, then the above $ ['(s)]^2 $ has to be replaced by $ s^2 $. 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 $ s=['(s)]=s $, then $ N(0, [s^2, ['(s)]^2])=N(0, s^2) $ 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 $ N([m, ['(m)]], [s^2, ['(s)]^2]) $-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.