Existence of Solutions for Stochastic Differential Equations under G-Brownian Motion with Discontinuous Coefficients

Abstract

The existence theory for the vector valued stochastic differential equations under G-Brownian motion (G-SDEs) of the type Xt = X0+ ∫t o(v;Xv)dv+ ∫t 0 g(v;Xv)d(B)v+ ∫t 0 h(v;Xv)dBv; t ∊ [0;T]; with first two discontinuous coefficients is established. It is shown that the G-SDEs have more than one solution if the coefficient g or the coefficients f and g simultaneously, are discontinuous functions. The upper and lower solutions method is used and examples are given to explain the theory and its importance.