  In the field of computational finance, it is common for the quantity of
interest to be expected values of functions of random variables via stochastic
differential equations (SDEs). For SDEs with globally Lipschitz coefficients
and commutative diffusion coefficients, the explicit Milstein scheme, relying
on only Brownian increments and thus easily implementable, can be combined with
the multilevel Monte Carlo (MLMC) method proposed by Giles
\cite{giles2008multilevel} to give the optimal overall computational cost
$\mathcal{O}(\epsilon^{-2})$, where $\epsilon$ is the required target accuracy.
For multi-dimensional SDEs that do not satisfy the commutativity condition, a
kind of one-half order truncated Milstein-type scheme without L\'evy areas is
introduced by Giles and Szpruch \cite{giles2014antithetic}, which combined with
the antithetic MLMC gives the optimal computational cost under globally
Lipschitz conditions. In the present work, we turn to SDEs with non-globally
Lipschitz continuous coefficients, for which a family of modified Milstein-type
schemes without L\'evy areas is proposed. The expected one-half order of strong
convergence is recovered in a non-globally Lipschitz setting, where the
diffusion coefficients are allowed to grow superlinearly. This helps us to
analyze the relevant variance of the multilevel estimator and the optimal
computational cost is finally achieved for the antithetic MLMC. The analysis of
both the convergence rate and the desired variance in the non-globally
Lipschitz setting is highly non-trivial and non-standard arguments are
developed to overcome some essential difficulties. Numerical experiments are
provided to confirm the theoretical findings.
