4 research outputs found
The truncated Milstein method for stochastic differential equations with commutative noise
Inspired by the truncated Euler-Maruyama method developed in Mao (J. Comput. Appl. Math. 2015), we propose the truncated Milstein method in this paper. The strong convergence rate is proved to be close to 1 for a class of highly non-linear stochastic differential equations with commutative noise. Numerical examples are given to illustrate the theoretical results
Strong convergence of an adaptive time-stepping Milstein method for SDEs with one-sided Lipschitz drift
We introduce explicit adaptive Milstein methods for stochastic differential
equations with one-sided Lipschitz drift and globally Lipschitz diffusion with
no commutativity condition. These methods rely on a class of path-bounded
timestepping 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
convergent of order one. This convergence 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.Comment: 20 pages, 2 figure