Existence and uniqueness of solutions for fractional stochastic delay differential equations

Abstract

The existence and uniqueness of solutions for a class fractional stochastic delay differential equations dx(t)=b(x(t), x(t-τ), t)dt+σ1(x(t), x(t-τ), t)dB(t)+σ2(x(t), x(t-τ), t)(dt)α of order α∈(0, 1) were studied. By introducing two new integral operators, the existence and uniqueness of solution of the equation was proved by using Picard successive approximation method and integral operator theory. The continuous dependence of the solution of the equation on the initial value was discussed.

Keywords

• fractional stochastic delay differential equations
• existence and uniqueness
• continuous dependence
• integral operators