The projected explicit Itô–Taylor methods for stochastic differential equations under locally Lipschitz conditions and polynomial growth conditions

A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

“…The classical (exponential) Euler approximations diverge in the strong and weak sense for most one-dimensional SDEs with super-linearly growing coefficients (see [25,27]) and also for some SPDEs (see Beccari et al [2]). It was shown in [26,24] that minor modifications of the Euler method -so called tamed Euler methods -avoid this divergence problem; see also the Euler-type methods, e.g., in [4,5,6,8,9,12,16,19,21,29,30,33,35,36,39,44,45,47,48]. Now, analogously to Hutzenthaler & Jentzen [23], Corollary 3.11 is a powerful tool to establish uniform strong convergence rates (in combination with exponential moment estimates for suitably tamed Euler approximations, e.g., Hutzenthaler et al [28]).…”

On the approximations of solutions to stochastic differential equations under polynomial condition

“…It should be stressed that, in the recent years, the existence and uniqueness of the exact solutions, development of the approximate methods, stability of the exact and approximate solutions and other qualitative and quantitative properties of the exact and approximate solutions, under highly nonlinear conditions on coefficients of the appropriate stochastic differential equations have attracted the attention of many researchers. We refer the reader, for example, to [4,8,17,20,21], among many other. So, the main aim of this paper is to provide a contribution to the analysis of stochastic differential equations with highly nonlinear drift coefficients, that is, with drifts which satisfy the polynomial condition.…”