Strongly asymptotically optimal schemes for the strong approximation of stochastic differential equations with respect to the supremum error

Abstract

Our subject of study is strong approximation of stochastic differential equations (SDEs) with respect to the supremum error criterion, and we seek approximations that are strongly asymptotically optimal in specific classes of approximations. We hereby focus on two principal types of classes, namely, the classes of approximations that are based only on the evaluation of the initial value and on at most finitely many sequential evaluations of the driving Brownian motion on average, and the classes of approximations that are based only on the evaluation of the initial value and on finitely many evaluations of the driving Brownian motion at equidistant sites. For SDEs with globally Lipschitz continuous coefficients, Müller-Gronbach [Ann. Appl. Probab. 12 (2002), no. 2, 664–690] showed that specific Euler–Maruyama schemes relating to adaptive and to equidistant time discretizations are strongly asymptotically optimal in these classes. In the present article, we generalize the results above to a significantly wider class of SDEs, such as ones with super-linearly growing coefficients. More precisely, we prove strong asymptotic optimality for specific coefficient-modified Euler–Maruyama schemes relating to adaptive and to equidistant time discretizations under rather mild assumptions on the underlying SDE. To illustrate our findings, we present two exemplary applications – namely, Euler–Maruyama schemes and tamed Euler schemes – and thereby analyze the SDE associated with the Heston-3∕2-model originating from mathematical finance.