Milstein-type methods for strong approximation of systems of SDEs with a discontinuous drift coefficient

We study strong approximation of d-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient driven by a d-dimensional Brownian motion W. More precisely, we essentially assume that the drift coefficient μ is piecewise Lipschitz continuous with an exceptional set Θ⊂ Rᵈ that is an orientable C⁵-hypersurface of positive reach, the diffusion coefficient σ is assumed to be Lipschitz continuous and, in a neighborhood of Θ, both coefficients are bounded and σ is non-degenerate. Furthermore, both μ and σ are assumed to be C¹ with intrinsic Lipschitz continuous derivative on Rᵈ Θ. We introduce, for the first time in literature, a Milstein-type method which can be used to approximate SDEs of this type for general d ∈ N and prove that this Milstein-type scheme achieves an Lₚ-error rate of order at least $3/4-$ in terms of the number of steps. This method depends, in addition to evaluations of W on a fixed grid, also on iterated integrals w.r.t. components of W, which can in general not be represented as functionals of W evaluated at finitely many time points. We additionally prove that our suggested Milstein-type method is only dependent on evaluations of W on a finite, fixed grid if σ is additionally commutative. To obtain our main result we prove that a quasi-Milstein scheme achieves an Lₚ-error rate of order at least $3/4-$ in our setting if μ is additionally continuous, which is of interest in itself.