Conditional G-expectation in $\mathbb{L}^{p}$ and related It\^o's calculus

TL;DR

This paper introduces a new form of conditional G-expectation within $ ext{L}^p$ spaces and develops an Itô calculus framework, including a generalized Itô's formula for smooth functions.

Contribution

It defines a dynamically consistent conditional G-expectation in $ ext{L}^p$ and establishes an associated stochastic calculus, extending Itô's formula to this setting.

Findings

Defined a new conditional G-expectation in $ ext{L}^p$.

Developed Itô's calculus for this expectation.

Proved Itô's formula for $C^{1,2}$ functions.

Abstract

In this paper, we define a dynamically consistent conditional G-expectation in space $\mathbb{L}^{p}$, and give the related stochastic calculus of It\^o's type, especially get It\^o's formula for a general $C^{1,2}$-function.