Faiz Faizullah

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

The existence theory for the vector valued stochastic differential equations under G-Brownian motion (G-SDEs) of the type Xt = X0+ ∫to(v;Xv)dv+ ∫t0 g(v;Xv)d(B)v+ ∫t0 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.

On the pth moment estimates of solutions to stochastic functional differential equations in the G-framework

The aim of the current paper is to present the path-wise and moment estimates for solutions to stochastic functional differential equations with non-linear growth condition in the framework of G-expectation and G-Brownian motion. Under the nonlinear growth condition, the pth moment estimates for solutions to SFDEs driven by G-Brownian motion are proved. The properties of G-expectations, Hölder’s inequality, Bihari’s inequality, Gronwall’s inequality and Burkholder–Davis–Gundy inequalities are used to develop the above mentioned theory. In addition, the path-wise asymptotic estimates and continuity of pth moment for the solutions to SFDEs in the G-framework, with non-linear growth condition are shown.

Existence of Solutions for G-SFDEs with Cauchy-Maruyama Approximation Scheme

We present the Cauchy-Maruyama (CM) approximation scheme and establish the existence theory of stochastic functional differential equations driven by G-Brownian motion (G-SFDEs). Several useful properties of Cauchy-Maruyama (CM) approximate solutions of G-SFDEs are given. We show that the unique solution of G-SFDEs gets convergence from Cauchy-Maruyama (CM) approximate solutions. The existence theorem for G-SFDEs is developed with the above mentioned scheme.

A Note on the Carathéodory Approximation Scheme for Stochastic Differential Equations under G-Brownian Motion

In this note, the Carathéodory approximation scheme for vector valued stochastic differential equations under G-Brownian motion (G-SDEs) is introduced. It is shown that the Carathéodory approximate solutions converge to the unique solution of the G-SDEs. The existence and uniqueness theorem for G-SDEs is established by using the stated method.

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,