MARTINGALE INEQUALITIES UNDER G-EXPECTATION AND THEIR APPLICATIONS

In this paper,we study the martingale inequalities under G-expectation and their applications.To this end,we introduce a new kind of random time,called G-stopping time,and then investigate the properties of a G-martingale(supermartingale)such as the optional sampling theorem and upcrossing inequalities.With the help of these properties,we can show the martingale convergence property under G-expectation.

Multiple Proxies on the Paleoenvironment of the Early Cambrian Marine Black Rock Series in the Tarim Basin, NW China: Molybdenum Isotope and Trace Element Evidence

The early Cambrian carbonaceous shale and laminated chert-phosphorite assemble (the black rock series) are widespread at the northwest margin of the Tarim Basin, Northwest China. In combination with previously reported data, we present stable molybdenum isotope (δ98/95Mo), TOC, and redox-sensitive trace elements to evaluate the sedimentary conditions in early Cambrian water column during the deposition of the black rock series in the Tarim Basin. Redox variation was documented based on enrichment factors (MoEF, VEF, and UEF) and redox-sensitiv elements ratios (Ni/Co, V/Cr, δU), etc. During the early Cambrian, there was transgressive event, and the sea level continues to rise. In the basal Cambrian, laminated chert-phosphorite assemble with low TOC concentrations suggest the oxic sedimentary condition in a restricted basin. Light Mo isotope values and redox sensitive elements enrichment in the carbonaceous shale layer indicate lack oxygenic sedimentary condition, and was suboxic/anoxic conditions during the transgressive phase. The hydrothermal fluids from the open ocean affected the whole deposition process of the black rock series.

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.