Caputo stochastic fractional differential equations: Carathéodory scheme and weak convergence

In this article, we consider Carathéodory scheme for scalar Caputo stochastic fractional differential equations (CSFDE) of order ν∈(78,1) in Lp spaces with p∈(22ν−1,11−ν) of the form (1) CD0+νy(t)=g(t,y(t))+h(t,y(t))dWtdt,t∈[0,T],(1) where T>0 is arbitrary, (Wt)t∈[0,T] denotes a standard Brownian motion on a completely filtered probability space (Ω,F,F:={Ft}t∈[0,T],P) and g, h:[0,T]×R→R are measurable functions. Based on the techniques of fractional calculus and Malliavin calculus, we establish an upper bound for |E[σ(yˆ(t))]−E[σ(y(t))]|, where y(t) represents the accurate solution of (Equation1(1) CD0+νy(t)=g(t,y(t))+h(t,y(t))dWtdt,t∈[0,T],(1) ) and yˆ(t) denotes the numerical solution of (Equation1(1) CD0+νy(t)=g(t,y(t))+h(t,y(t))dWtdt,t∈[0,T],(1) ), as defined by (Equation5(5) yˆ(t)=y0+1Γ(ν)∫0tg(ς,yˆ(ς−Δ))(t−ς)1−νdς+1Γ(ν)∫0th(ς,yˆ(ς−Δ))(λn(t)−γn(ς))1−νdWς,(5) ) below, with σ∈B. Here, B is defined as follows: B:={σ:R→Rismeasurablefunction:∫−∞+∞|σ(x)|dx