A note on convergence and stability of the truncated Milstein method for stochastic differential equations

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

“…The following corollary, Corollary 3.2, provides marginal and uniform Lyapunov-type estimates for solutions of SDEs. The marginal Lyapunov-type estimate (44) To the best of our knowledge, the uniform Lyapunov-type esimate (45) below is new. In the literature, uniform moment estimates are derived with the help of a Burkholder-Davis-Gundy inequality.…”

“…for the 3/2-model of Heston [20] and Platen [38] or for the 4/2-model of Grasselli [14]. In many situations where upper bounds for uniform moments could be established, these are less sharp than (45); see, e.g., Proposition 2.27 in Cox et al [10] (with V depending only on the first component). Corollary 3.2 follows directly from Corollary 2.5 (applied with a s = µ(s, X s ), b s = σ(s, X s ), α s = α, β s = β, q 2 = ∞ for all s ∈ [0, T ] in the notation of Corollary 2.5).…”

“…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]).…”