On tamed milstein schemes of SDEs driven by Lévy noise

Abstract

We extend the taming techniques developed in [3,19] to construct explicit Milstein schemes that numerically approximate Lévy driven stochastic differential equations with super-linearly growing drift coefficients. The classical rate of convergence is recovered when the first derivative of the drift coefficient satisfies a polynomial Lipschitz condition.

Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations

“…Fully drift-implicit Euler schemes can, however, often only be simulated approximatively as a nonlinear equation has to be solved in each time step and the resulting approximations of the fully drift-implicit Euler approximations require additional computational effort (particularly, when the state space of the considered SEE is high dimensional, see, e.g., Figure 4 in Hutzenthaler et al [20]) and have not yet been shown to converge strongly. Recently, a series of explicit and easily implementable timediscrete approximation schemes have been proposed and shown to converge strongly in the case of SEEs with superlinearly growing nonlinearities; see, e.g., Hutzenthaler et al [20], Wang & Gan [45], Hutzenthaler & Jentzen [18], Tretyakov & Zhang [44], Halidias [13], Sabanis [39,40], Halidias & Stamatiou [15], Hutzenthaler et al [22], Szpruch & Zhāng [42], Halidias [14], Liu & Mao [30], Hutzenthaler & Jentzen [17], Zhang [46], Dareiotis et al [10], Kumar & Sabanis [27], Beyn et al [1], Zong et al [47], Song et al [41], Ngo & Luong [36], Tambue & Mukam [43], Mao [34], Beyn et al [2], Kumar & Sabanis [28], and Mao [35] in the case of finite dimensional SEEs and see, e.g., Gyöngy et al [12] and Jentzen & Pušnik [25] in the case of infinte dimensional SEEs. These schemes are suitable modified versions of the explicit Euler scheme that somehow tame/truncate the superlinearly growing nonlinearities of the considered SEE and thereby prevent the considered tamed scheme from strong divergence.…”

“…These schemes are suitable modified versions of the explicit Euler scheme that somehow tame/truncate the superlinearly growing nonlinearities of the considered SEE and thereby prevent the considered tamed scheme from strong divergence. However, each of the above mentioned temporal strong convergence results for implicit (see [16,11,26]) and explicit (see [20,45,18,44,13,39,40,15,22,42,14,30,17,46,10,27,12,1,47,41,25,36,43,34,2,28,35]) schemes applies merely to trace noise class driven SEEs and excludes the important case of the more irregular space-time white noise. In particular, none of these results applies to space-time white noise driven stochastic Ginzburg-Landau equations.…”

Convergence rate of the truncated Milstein method of stochastic differential delay equations

“…The proof for the method was modified in [18]. The tamed Milstein methods were developed in [19] and [8] for SDEs driven by Brownian motion and Lévy noise, respectively.…”

Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

“…This allows to prove the strong convergence rate 1 in the case of SODEs whose drift coefficient functions satisfy a one-sided Lipschitz condition. The same approach is used in [12], where the authors consider SODEs driven by Lèvy noise. However, both papers still require that the diffusion coefficient functions are globally Lipschitz continuous.…”