Convex Order, Quantization and Monotone Approximations of ARCH Models

We are interested in proposing approximations of a sequence probability measures the convex order by finitely supported still order. propose to alternate two types operators: transition according one-step martingale Markov kernel mapping measure its successor and spatial discretization through dual (also called Delaunay) quantization. In case autoregressive conditional heteroskedasticity (ARCH) models particular Euler scheme driftless Brownian diffusion, noise has be truncated enable quantization step. analyze error between original ARCH model approximation with exhibit conditions under which latter is dominated former at level sample paths. Last, we combining steps truncation primal