Qualitative properties of numerical methods for the inhomogeneous geometric Brownian motion
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We introduce the inhomogeneous geometric Brownian motion (IGBM) as a test equation for analysing qualitative features of numerical methods applied to multiplicative noise stochastic ordinary differential equations of Ito type with an inhomogeneous drift. The usual linear stability analysis of a constant equilibrium (in the mean-square or almost-sure sense) cannot be carried out for these equations as they do not have one. However, the conditional and asymptotic mean and variance of the IGBM are explicitly known and the process can be characterised according to Feller's boundary classification. We compare four splitting schemes, two based on the Lie-Trotter composition and two based on the Strang approach, with the frequently used Euler-Maruyama and Milstein methods. First, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered numerical schemes and analyse the resulting errors with respect to the true quantities. In contrast to the frequently applied methods, the splitting schemes do not require extra conditions for the existence of the asymptotic moments and outperform them in terms of the variance. Second, we prove that the constructed splitting schemes preserve the boundary properties of the process, independently of the choice of the discretisation step, whereas the Euler-Maruyama and Milstein methods do not.
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