Convergence rates for adaptive approximation of ordinary differential equations

Convergence Rates for Adaptive Weak Approximation of Stochastic Differential Equations

“…The difference between the two algorithms is that the stochastic time steps may use different meshes for each realization, based on successive Brownian bridge sampling of a Brownian motion realization, while the deterministic time steps use the same mesh for all realizations of the Brownian motion W = W 1 W 0 . The construction and the analysis of the adaptive algorithms are inspired by the related work of Moon et al [32], on adaptive algorithms for deterministic ordinary differential equations, and the error estimates from Szepessy et al [39].…”

“…Based on Morin et al [33], the work of Binev et al [7] and Stevenson [38] extend the analysis of Cohen et al [10] to include finite element approximation. The work of Moon et al [32] connects DeVore's smoothness conditions to error densities for adaptive a.s. convergence of the approximate solution, X, as the maximal step size tends to zero. Although the time steps are not adapted to the standard filtration generated by only W for the stochastic time stepping algorithm, the work of Szepessy et al [39] proved that the corresponding approximate solution converges to the correct adapted solution X.…”

“…Although the work Szepessy et al [39] proposed adaptive algorithms, the main focus in that work was on error estimates. Properties regarding the stopping, efficiency, and accuracy of the adaptive algorithms, following the ideas in Moon et al [32] are first studied here. Assume that the process X satisfies (1) and its approximation, X, is given by (2), then the error expansions in Theorems 1.2 and 2.2 of Szepessy et al [39] have the form…”

“…To achieve (31) for each realization, start with an intial partition t 1 and then specify iteratively a new partition t k + 1 , from t k , using the following refinement strategy: For each realization in the mth batch, for each time step n = 1 2 N k if r n k ≤ s 1 TOL T N M − 1 then divide t n k into H = 2 uniform substeps else let the new step be the same as the old endif endfor (32) The refinement strategy (32) motivates the following stopping criteria: For each realization of the mth batch,…”

“…Suppose the adaptive algorithm uses the strategy (32) and (33). Assume that c satisfies (36) for the time steps corresponding to the maximal error indicator on each refinement level, and that…”