Harnack Inequality and Gradient Estimate for $G$-SDEs with Degenerate Noise

Abstract

In this paper, the Harnack inequalities for $G$-SDEs with degenerate noise are derived by method of coupling by change of measure. Moreover, the gradient estimate for the associated nonlinear semigroup ${\bar{P}}_t$ $$|\nabla {\bar{P}}_{t}f|\le c(p,t)({\bar{P}}_t|f{|}^p{)}^{\frac{1}{p}},p>1,t>0$$ is also obtained for bounded and continuous function $f$. As an application of Harnack inequality, we prove the weak existence of degenerate $G$-SDEs under some integrable conditions. Finally, an example is presented. All of the above results extends the existed results in the linear expectation setting.

17 pages

References in corpus (4)

• A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
• Girsanov's formula for G-Brownian motion
• Dynamic Programming Principle for Stochastic Recursive Optimal Control Problem under G-framework
• Harnack Inequality and Applications for SDEs Driven by $G$-Brownian motion

Cited by in corpus (1)

• Harnack and log Harnack Inequalities for $G$-SDEs with Multiplicative Noise