Existence of solutions for G-SDEs with upper and lower solutions in the reverse order

Motivated from the risk measures, superhedging in finance and uncertainties in statistics, the G-Brownian motion was introduced by Peng (2006). The related stochastic calculus in the framework of a sublinear expectation (known as G-expectation) is developed (Peng, 2006, 2008). He introduced the stochastic differential equations driven by G-Brownian motion (GSDEs) and established the existence and uniqueness of solutions for G-SDEs with Lipschitz continuity condition on the coefficients (Peng, 2006, 2008). The G-SDEs with integral Lipschitz conditions were studied in Bai and Lin (2010) and with global Carathéodory conditions in Ren and Hu (2011) and Gao (2009). In contrast to the aforementioned, here the existence theory for G-SDEs whose drift coefficients are discontinuous functions is developed by the method of upper and lower solutions in the reverse order. The importance of discontinuous functions is not uncommon. For example, the unit step function or the Heaviside function R R → : H , defined by,