Sublinear Expectations: On Large Sample Behaviours, Monte Carlo Method, and Coherent Upper Previsions

Abstract

Shige Peng's sublinear expectations generalize ordinary linear expectation operators. It is shown that the behaviour of sample averages of Peng i.i.d. variables may be very different from the probabilistic intuition. In particular, Peng's generalization of the Monte Carlo method is shown to be wrong. It is also observed that sublinear expectations coincide with Peter Walley's coherent upper previsions on a linear space.

Generalized Feynman-Kac Formula under volatility uncertainty

“…Note that, as τ is quasicontinuous, it follows by Corollary 2.17 that 2 and by using the same arguments as in (4. 19)-(4.20) we get by (4.28) that…”

“…By using Theorem 4.11 we are able to numerically approximate the G-conditional expectation in (1.1) in a computationally efficient way without resolving to Monte Carlo methods. This is also a further contribution since it is well known that the strong law of large numbers in the G-setting differs from the one in the classical framework, see Theorem 3.1, Chapter II in [16], and this may cause some issues in the application of Monte-Carlo algorithms, see for example [19].…”

Exponential inequalities and a strong law of large numbers for END random variables under sub-linear expectations

“…Terán [19] showed that (2.2) and (2.5) don't hold in general by a counterexample in which µ = E[X 1 ] = 0, µ = E[X 1 ] = 1 and S n (ω)/n → 0 for all ω ∈ Ω. Zhang [21] pointed out that, under the framework of sub-linear expectation, the continuity of the capacity is a very strict condition by showing that, if a capacity V which satisfies (1.5) is continuous and there is a sequence…”

“…For (2.8), the condition (CC) is not needed. The counterexample given by Terán [19] shows that (2.9) and (2.10) may fail if the condition (CC) is not satisfied (c.f. Theorem 3.1 of [19]).…”

Stein’s method for the law of large numbers under sublinear expectations

“…Example 5.2 Let Ω be the same space as is defined in Example 5.1, and let Ω 0 be the subset of Ω with ξ i = 0 for all except a finite number i ∈ N. i) ( [16])…”