Hyperfinite Construction of $G$-expectation

TL;DR

This paper introduces a hyperfinite discrete analogue of $G$-expectation using nonstandard analysis, establishing foundational theory and proving an existence theorem for liftings of continuous-time $G$-expectation.

Contribution

It develops the basic theory of hyperfinite $G$-expectations and proves a new existence theorem for liftings, connecting discrete and continuous $G$-expectations.

Findings

Established a hyperfinite $G$-expectation framework.

Proved an existence theorem for liftings of continuous-time $G$-expectation.

Developed a new discretization theorem for $G$-expectation.

Abstract

The hyperfinite $G$-expectation is a nonstandard discrete analogue of $G$-expectation (in the sense of Robinsonian nonstandard analysis). A lifting of a continuous-time $G$-expectation operator is defined as a hyperfinite $G$-expectation which is infinitely close, in the sense of nonstandard topology, to the continuous-time $G$-expectation. We develop the basic theory for hyperfinite $G$-expectations and prove an existence theorem for liftings of (continuous-time) $G$-expectation. For the proof of the lifting theorem, we use a new discretization theorem for the $G$-expectation (also established in this paper, based on the work of Dolinsky et al. [Weak approximation of $G$-expectations, Stoch. Proc. Appl. 122(2), (2012), pp.664--675]).