Eur. Phys. J. Plus (2017) 132: 435
https://doi.org/10.1140/epjp/i2017-11700-9

Regular Article

On existence and approximate solutions for stochastic differential equations in the framework of G-Brownian motion

¹ School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, Shandong, China
² Department of BS & H, College of E & ME, National University of Sciences and Technology (NUST), Rawalpindi, Pakistan

^* e-mail: faiz_math@ceme.nust.edu.pk

Received: 29 April 2017
Accepted: 11 September 2017
Published online: 20 October 2017

Abstract

This investigation aims at studying a Euler-Maruyama (EM) approximate solutions scheme for stochastic differential equations (SDEs) in the framework of G-Brownian motion. Subject to the growth condition, it is shown that the EM solutions are bounded, in particular, . Letting Z(t) as a unique solution to SDEs in the G-framework and utilizing the growth and Lipschitz conditions, the convergence of to Z(t) is revealed. The Burkholder-Davis-Gundy (BDG) inequalities, Hölder’s inequality, Gronwall’s inequality and Doobs martingale’s inequality are used to derive the results. In addition, without assuming a solution of the stated SDE, we have shown that the Euler-Maruyama approximation sequence is Cauchy in thus converges to a limit which is a unique solution to SDE in the G-framework.