Adaptive weak approximation of stochastic differential equations

Anders Szepessy*, Raúl Tempone, Georgios E. Zouraris

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

52 Scopus citations


Adaptive time-stepping methods based on the Monte Carlo Euler method for weak approximation of Itô stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading-order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or deterministic time steps are described. Numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps. Stochastic time steps use Brownian bridges and require more work for a given number of time steps. Deterministic time steps may yield more time steps but require less work; for example, in the limit of vanishing error tolerance, the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps.

Original languageEnglish (US)
Pages (from-to)1169-1214
Number of pages46
JournalCommunications on Pure and Applied Mathematics
Issue number10
StatePublished - Oct 2001
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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