Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations

By Desmond John Higham, Xuerong Mao, Andrew M. Stuart. SIAM Journal on Numerical Analysis, 2002

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Subfields: Computational Mechanics, Finance, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5d4ad2f47c4e2acaec807c21334430068ec533bb

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Citations in the corpus

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients (2013, background): “1) are studied but rates of convergence are not considered and a weaker measure of error is taken, and [16], where rates are proved for SDEs with globally Lipschitz diffusion term.”
The truncated Euler-Maruyama method for stochastic differential equations (2015, methodology): “Instead of imposing these known conditions, Higham, Mao and Stuart [10] proposed the bounded condition on the pth moments of both exact solution and numerical solution to the underlying SDE and proved the strong convergence theory.”
Proximal Markov chain Monte Carlo algorithms (2013, methodology): “Theorem 3.4 Suppose that π ∈ C 1 and that (1) holds.”
The Numerical Approximation of Stochastic Partial Differential Equations (2009, methodology): “9) was handled by Higham et al [56] and Mattingly et al [94] using dissipativity and ergodic arguments.”
Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process (2013, methodology): “The drift implicit Euler scheme (also known as backward Euler scheme) has been studied by Higham, Mao and Stuart [4] for SDEs on R with a Lipschitz condition on the diffusion coefficient and a monotonicity condition on the drift coefficient that extends (5).”
The Stochastic Model (2018, background): “2/5 their values at positive equilibrium E∗ [13-15].”
Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model (2011, methodology): “} of the time interval [0, ∞), and introduce the backward Euler-Maruyama (BEuM) method [ 14 , 17]”
A numerical scheme for stochastic PDEs with Gevrey regularity (2004, positive): “Next the noise contribution is analysed: using Doob’s martingale inequality [15] as in [9], we get E  sup t∈(0,τ ] ∑ |n|≤N (1 + n) ∣∣∣∣∣ [t/h] ∑”
Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations (2013, neutral): “, [20, 18, 22, 33, 25] and the references mentioned therein.”
Characterization of bistability for stochastic multistep methods (2010, neutral): “In [16] the authors consider Euler-Maruyama type schemes under onesided Lipschitz and polynomial growth conditions on the drift b0 and estimates on the higher moments of the solution.”
Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift (2016, positive): “2 in [10]), these solutions are stable in the sense that for any T, p > 0, E [ sup0≤t≤T ‖Xt‖ ] < ∞.”
Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment, and stability (2019, methodology): “(3.37) On the other hand, for each ∈ (0, ∗], we apply the EM method to SDE (3.36) and we denote by u(t) the piecewise constant EM solution (see Kloeden & Platen, 1992; Higham et al., 2002) that has the property E ( sup 0≤t≤T ∣ ∣y(t) − u(t)∣∣q ) ≤ C q/2 ∀ T ≥ 0.”
Loss of regularity for Kolmogorov equations (2012, background): “3 in Higham, Mao & Stuart [29], Schurz [66], Theorems 2 and 3 in Higham & Kloeden [28], Theorem 6.”