Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients

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.…”

Equivalence of the mean square stability between the partially truncated Euler–Maruyama method and stochastic differential equations with super-linear growing coefficients

“…However, due to the advantages such as simple structure and less computational cost in each iteration (not like implicit methods in which some non-linear equation system needs to be solved in each iteration) [8], the explicit Euler-type methods are still attracting lots of attention. Therefore, in the past several years some modified explicit Euler methods, such as the tamed Euler method [13,24,25,29] and the truncated EM method [6,15,16,20,21], have been developed. The bloom of explicit methods brings the second motivation of this paper as follows.…”

“…More precisely, we prove that (a) the mean square stability of the SDE implies the mean square stability of the partially truncated EM method, (b) the mean square stability of the partially truncated EM method implies the mean square stability of the SDE if the step size of the numerical method is carefully chosen. To our best knowledge, few works have dealt with (B) using explicit methods for SDEs with super-linear coefficients, though some works have studied (A) using explicit methods [6,29]. As pointed out in [9], sometimes (B) is more interesting and important since if (B) is true for some numerical method, then by carefully conducting the numerical simulation one can know whether the SDE is stable or not without using the Lyapunov technique.…”

Stochastic symplectic partitioned Runge–Kutta methods for stochastic Hamiltonian systems with multiplicative noise

“…Most SDEs arising in practice are nonlinear, and cannot be solved explicitly, so the construction of efficient numerical methods is of great importance. In the recent years many numerical methods for SDEs have been designed, for example see [3][4][5][6][7][8][9][10][11].…”