Euler–Maruyama schemes for Caputo stochastic fractional delay differential equations

Abstract The main aim of this paper is to develop Euler–Maruyama type schemes for Caputo stochastic fractional delay differential equations (Caputo SFDDEs) of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \alpha\in(\frac{1}{2},1) . Under the standard Lipschitz and Hölder continuity conditions on the coefficients, the strong convergence and convergence rate of these schemes are established. Specifically, in the case that the coefficients of the Caputo SFDDEs are time-invariant, the convergence rate is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>−</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:math> \alpha-\frac{1}{2} . A numerical example is provided to illustrate the theoretical results.