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

Abstract We introduce an explicit adaptive Milstein method for stochastic differential equations with no commutativity condition. The drift and diffusion are separately locally Lipschitz together satisfy a monotone This relies on class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach boundary sphere, invoking backstop in event that timestep becomes too small. prove such schemes strongly $$L_2$$ L 2 convergent order one. is inherited Euler–Maruyama scheme additive noise case. Moreover we show probability using at any step can be made arbitrarily compare our to other fixed-step variants range test problems.