Convergence of diffusions and their discretizations: from continuous to discrete processes and back

Understand

In this paper, we establish new quantitative convergence bounds for a class of functional autoregressive models in weighted total variation metrics.

• To derive our results, we show that under mild assumptions, explicit minorization and Foster-Lyapunov drift conditions hold.
• The main applications and consequences of the bounds we obtain concern the geometric convergence of Euler-Maruyama discretizations of diffusions with identity covariance matrix.
• Second, as a corollary, we provide a new approach to establish quantitative convergence of these diffusion processes by applying our conclusions in the discrete-time setting to a well-suited sequence of discretizations whose associated stepsizes decrease towards zero.