Numerical Solution of Stochastic Differential Equations

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Atomistic spin model simulations of magnetic nanomaterials (2013, methodology): “Although the Heun scheme was derived specifically for a stochastic equation with multiplicative noise, in the absence of the noise term the Heun method reduces to a standard second order Runge–Kutta method [120].”
Second order discretization of backward SDEs and simulation with the cubature method (2014, background): “2 of Chapter 5 of Kloeden and Platen [16].”
Langevin diffusions and the Metropolis-adjusted Langevin algorithm (2013, methodology): “The Euler–Maruyama method (e.g. Kloeden and Platen, 1992) is among the simplest: for a chosen step-size h a realisation the sequence of random variables Xh,X2h, ...,Xnh is approximated using the procedure x(i+1)h = xih + hb(xih) + σ(xih)ε, where ε ∼ Nd(0, hId×d).”
RUNGE-KUTTA SCHEMES FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS (2014, background): “7 in [9]), we then compute that k0(β) + k1(β) =m+1 and k0((j)) = k1((j)) = 0.”
Loss of regularity for Kolmogorov equations (2012, background): “, Kloeden and Platen [42]) the Euler–Maruyama scheme is shown to converge in the numerically weak sense with order 1 if the coefficients of the SDE are smooth and globally Lipschitz continuous (see Chapter 8 in Kloeden and Platen [42] for the precise assumptions) and, therefore, the order line 1 is ”