非齐次几何布朗运动数值方法的定性性质
Qualitative properties of numerical methods for the inhomogeneous geometric Brownian motion
摘要：
对非齐次几何布朗运动的不同数值方法的定性特征进行了比较分析。IGBM的条件和渐近均值和方差是已知的，并且该过程可以根据Feller的边界分类来表征。我们比较常用的Euler-Maruyama和Milstein方法，两个Lie-Trotter和两个Strang分裂计划和两个方法的基础上的常微分方程（ODE）的方法，即经典的Wong-Zakai近似和最近提出的对数ODE计划。首先，我们证明，在对比欧拉-丸山和米尔斯坦计划，分裂和ODE计划保持边界属性的过程中，独立的时间离散化步骤的选择。其次，我们推导出封闭形式的表达式的条件和渐近的均值和方差的所有考虑计划，并分析由此产生的偏差。虽然Euler-Maruyama和Milstein方案是唯一可能具有渐近无偏均值的方法，但分裂和ODE方案在方差保持方面表现得更好。Strang格式优于Lie-Trotter分裂，log-ODE格式优于经典ODE方法。对数常微分方程方案的均值和方差偏差对于许多相关的参数设置是非常小的。然而，在某些情况下，两个派生的斯特朗分裂可能是一个更好的选择，其中一个需要相当少的计算工作比对数常微分方程方法。所提出的分析可以以类似的方式进行其他数值方法和随机微分方程具有可比的功能。
英文摘要：
We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The conditional and asymptotic mean and variance of the IGBM are known and the process can be characterised according to Feller's boundary classification. We compare the frequently used Euler-Maruyama and Milstein methods, two Lie-Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong-Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler-Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler-Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie-Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features.
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