Continuous dependence of the solution of a stochastic differential equation with nonlocal conditions

A. M. A. El-Sayed a,∗ , R. O. Abd-El-Rahman b and M. El-Gendy c

Continuous dependence of the solution of a stochastic differential equation with nonlocal conditions

Authors :

A. M. A. El-Sayed a^,∗, R. O. Abd-El-Rahman ^b and M. El-Gendy ^c

Author Address :

^a,b,cDepartment of Mathematics, College of science, Alexandria university, Egypt.

*Corresponding author.

Abstract :

In this paper we are concerned with a nonlocal problem of a stochastic differential equation that contains a Brownian motion. The solution contains both of mean square Riemann and mean square
Riemann-Steltjes integrals, so we study an existence theorem for unique mean square continuous solution and its continuous dependence of the random data $X_0$ and the (non-random data) coefficients of the nonlocal condition $a_k$. Also, a stochastic differential equation with the integral condition will be considered.

Keywords :

Integral condition, Brownian motion, unique mean square solution, continuous dependence, random data, non-random data, integral condition.

DOI :

Article Info :

Received : January 10, 2016; Accepted : July 21, 2016.