Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

Cónall Kelly, Gabriel J. Lord, Fandi Sun*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
24 Downloads (Pure)

Abstract

We introduce an explicit adaptive Milstein method for stochastic differential equations with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly L2 convergent of order one. This order is inherited by an explicit adaptive Euler–Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.
Original languageEnglish
Article number33
JournalBIT Numerical Mathematics
Volume63
Issue number2
Early online date22 May 2023
DOIs
Publication statusPublished - Jun 2023

Keywords

  • Strong convergence
  • 65C30
  • 60H35
  • Stochastic differential equations
  • Adaptive time-stepping
  • Non-globally Lipschitz coefficients
  • Milstein method
  • 60H10

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