Existence, uniqueness and stability of the solution to neutral stochastic functional differential equations with infinite delay under non-Lipschitz conditions

Abstract

Choosing space C g as the phase space, the existence, uniqueness and stability of the solution to neutral stochastic functional differential equations with infinite delay (short for INSFDEs) are studied in this paper. Under non-Lipschitz condition, weakened linear growth condition and contractive condition, the existence-and-uniqueness theorem of the solution to INSFDEs by means of the Picard iteration, Doob's martingale inequalities, Gronwall's inequality and Bihari's inequality is obtained. Furthermore, the continuous dependence of the solutions on the initial value to INSFDEs are derived. MSC: 65C30; 60H10

Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

“…In the present paper, we proved a type of existence and uniqueness theorem of a solution of the neutral stochastic differential equation using the weakened conditions when the conditions are in the form of (4)- (6). Our main result does not cover the more general case of existence and uniqueness of the stochastic equation under some weakened conditions.…”

“…A more realistic model would include some of the past and present values, but that involves derivatives with delays as well as the function of the system. These equations have historically been referred to as neutral stochastic functional differential equations, or neutral stochastic differential delay equations [1][2][3][4][5][6].…”

“…After that, the study of the existence and uniqueness theorem for stochastic differential equations (SDEs) and NSDEs developed into some new uniqueness theorems for SDEs and NSDEs under special conditions. See [1,2,6,[8][9][10][11][12][13][14][15], and references therein for details.…”

Caratheodory's approximate solution to stochastic differential delay equation

“…n } stands for the Lebesgue measure on R. For some more details on stochastic differential equations, refer to [1][2][3][5][6][7][8][9][10][11] and references therein. By using the nonlinear growth condition and nonlinear growth condition, in 2015, Kim [4] studied the difference between the approximate solution and the accurate solution to the stochastic differential delay equation (shortly, SDEs).…”

Carathéodory’s approximate solution to stochastic differential delay equation

“…For results related to the stochastic differential delay equation, see [2]- [4], [6]- [10], [12], [13], and references therein for details.…”