On a Coupled System of Stochastic It<inline-formula><math display="inline"><semantics><mover accent="true"><mi>o</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula>-Differential and the Arbitrary (Fractional) Order Differential Equations with Nonlocal Random and Stochastic Integral Conditions

Abstract

The fractional stochastic differential equations had many applications in interpreting many events and phenomena of life, and the nonlocal conditions describe numerous problems in physics and finance. Here, we are concerned with the combination between the three senses of derivatives, the stochastic Ito^-differential and the fractional and integer orders derivative for the second order stochastic process in two nonlocal problems of a coupled system of two random and stochastic differential equations with two nonlocal stochastic and random integral conditions and a coupled system of two stochastic and random integral conditions. We study the existence of mean square continuous solutions of these two nonlocal problems by using the Schauder fixed point theorem. We discuss the sufficient conditions and the continuous dependence for the unique solution.

Keywords

• stochastic processes
• It<i>ô</i>-differential equations
• random differential equations
• stochastic differential equation
• coupled system
• fractional order derivative