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Z. Naturforsch. 67a, 692 – 698
(2012)
doi:10.5560/ZNA.2012-0092
Existence of Solutions for Stochastic Differential Equations under G-Brownian Motion with Discontinuous Coefficients
Department of Basic Sciences and Humanities, College of Electrical and Mechanical Engineering (EME), National University of Sciences and Technology (NUST), Islamabad, Pakistan and
College of Physical and Environmental Oceanography, Ocean University of China, Qingdao 266100, PR China
Received May 15, 2012 / revised August 16, 2012 / published online November 28, 2012
Reprint requests to: F. F.; E-mail:
faiz_math@yahoo.com
The existence theory for the vector valued stochastic differential equations under G-Brownian motion (G-SDEs) of the type Xt = X0 +∫0tf(v, Xv)dv +∫0tg(v, Xv)d〈B〉v +∫0th(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.
Key words: Stochastic Differential Equations; G-Brownian Motion; Discontinuous Coefficients; Existence; Upper and Lower Solutions.
Mathematics Subject Classification 2000: 60H10; 60H20