Extension and Application of Itô's Formula Under G-Framework

Abstract

In continuing his study of the intrinsically nonlinear expectation and conditional expectation under the so-called G-framework, Peng introduced a nonlinear Itô calculus; here, the G refers to the generator of a nonlinear heat equation. There, he derived the corresponding Itô formula for C ²-functions with bounded Lipschiz derivatives. This restrictive class of functions limits its applicatory value to stochastic finances and cannot be applied to study the powers of the G-Brownian motion. We extend the Itô formula to a slightly more general class of functions (C ²-functions with uniformly continuous derivatives). This enables us to compute the G-expectations of the even powers of the G-Brownian motion. The G-expectation of odd powers behave differently; in particular, we show that the G-expectation of the cube of the G-Brownian motion is positive, which is qualitatively different from the classical Brownian motion case. We remark that we are not able to get a formula for the G-expectation of the general odd powers of the G-Brownian motion.

Keywords:

• Backward stochastic differential equation
• Non linear expectation
• Itô formula

Mathematics Subject Classification:

• 60F15
• 60H30
• 62F12

This research was supported in part by the National Natural Science Foundation of China under grant nos. 10771214 and 10901168, and in part by the Project of Chinese Ministry of Education, 09YJCZH122.