Word combinatorics for stochastic differential equations: splitting integrators Articles uri icon

publication date

  • July 2019

start page

  • 2163

end page

  • 2195


  • 4


  • 18

International Standard Serial Number (ISSN)

  • 1534-0392

Electronic International Standard Serial Number (EISSN)

  • 1553-5258


  • We present an analysis based on word combinatorics of splitting integrators for Ito or Stratonovich systems of stochastic differential equations. In particular we present a technique to write down systematically the expansion of the local error; this makes it possible to easily formulate the conditions that guarantee that a given integrator achieves a prescribed strong or weak order. This approach bypasses the need to use the Baker-Campbell-Hausdorff (BCH) formula and shows the existence of an order barrier of two for the attainable weak order. The paper also provides a succinct introduction to the combinatorics of words.


  • Mathematics
  • Physics
  • Robotics and Industrial Informatics


  • stochastic differential equations; splitting integrators; word combinatorics; numerical-integration; formal series; dynamical-systems; invariants; algebra; trees