Local time and Tanaka formula of $G$-martingales

TL;DR

This paper develops the theory of local time and Tanaka formula for symmetric G-martingales within the G-expectation framework, extending classical stochastic calculus to nonlinear expectations.

Contribution

It introduces the local time for G-martingales, proves its properties, and establishes a Tanaka formula for convex functions of G-martingales, advancing nonlinear stochastic analysis.

Findings

Local time of G-martingales belongs to G-L2 space.

Existence of a bicontinuous modification of local time.

Tanaka formula for convex functions of G-martingales established.

Abstract

The objective of this paper is to study the local time and Tanaka formula of symmetric $G$-martingales. We introduce the local time of $G$-martingales and show that they belong to $G$-expectation space $L_{G}^{2}(\Omega _{T})$. The bicontinuous modification of local time is obtained. We finally give the Tanaka formula for convex functions of $G$-martingales.