Numerical simulation of a linear stochastic oscillator with additive noise
Consider ordinary stochastic differential equations (linear oscillators) governed by \[ dX(t) = Y(t) dt \] \[ dY(t) = - X(t) dt + \sigma dW(t) \] perturbed by additive noise, where \(W(t), t \geq 0\) is a standard \(d\)-dimensional Wiener process (\(W(0)=0\), \(E [W(t)]=0\), \(E [W(t)]^2=t\)), and \(\sigma\) is a nonrandom real-valued constant as its noise intensity. Assume that \(X(0)=x_0=1\) and \(Y(0)=0\) are fixed throughout the paper. Suppose that one discretizes this type of equation with constants \(\sigma > 0\) by Euler-type methods along nonrandom equidistant partitions of time-intervals \([0,T]\), where \(\Delta\) is the nonrandom mesh size. The ability of such numerical methods to reproduce long-time features of the above stated linear stochastic oscillator is examined. It is shown that the standard forward Euler-Maruyama method fails to capture the correct second moment growth rate, whereas a customized extension of the partitioned Euler method behaves well in this respect. Such a qualitative failure of standard Euler-Maruyama methods applied to more general systems of linear equations with both additive and multiplicative noise has been demonstrated already in the book of \textit{H. Schurz} [Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications, (Logos-Verlag, Berlin) (1997; Zbl 0905.60002)]; and the paper of \textit{H. Schurz} [ZAMM, Z. Angew. Math. Mech. 79, No. 6, 375--382 (1999; Zbl 0938.34072)] among other publications. Thus, the main new result of this paper can be seen that the authors show that the partitioned Euler method inherits an infinite oscillation property. A weaker oscillation result is also proved for a wide class of Itô-Taylor methods applied to this linear oscillator with \(x_0=1\) and \(y_0=0\).
scientific article; zbMATH DE number 2107329