Convergence rate of strong approximations of compound random maps

We consider a random map x → F (ω, x) and a random variable Θ(ω), and we denote by F^N (ω, x) and Θ^N (ω) their approximations: We establish a strong convergence result, in Lp-norms, of the compound approximation F^N (ω, Θ^N (ω)) to the compound variable F (ω, Θ(ω)), in terms of the approximations of F and Θ. We then apply this result to the composition of two Stochastic Differential Equations through their initial conditions, which can give a way to solve some Stochastic Partial Differential Equations