Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

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.