自适应性在Cox-Ingersoll-Ross模型数值方法中的作用
The role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model
摘要：
我们证明了自适应显式欧拉方法的Cox-Ingersoll-Ross模型的近似解的有效性。这依赖于一类路径有界的时间步长策略，其工作原理是当解决方案接近零的邻域时减小步长。该方法是混合动力的意义上说，一个收敛的backstop方法调用，如果时间步长变得太小，或防止解决方案过冲零，成为负的。在隐含Feller条件的参数约束下，我们证明了这样的方案是强收敛的，至少为1/2阶。强误差的控制对于多水平蒙特卡罗技术是重要的。在Feller条件下，我们还证明了需要backstop方法来防止负值的概率可以任意小。数值上，我们比较了这种自适应方法固定步长隐式和显式计划，和一个新的半隐式自适应变体。我们观察到，自适应的方法导致的方法，是有竞争力的领域，超出费勒的条件，表明适合的建模随机波动率在赫斯顿型资产模型。
英文摘要：
We demonstrate the effectiveness of an adaptive explicit Euler method for the approximate solution of the Cox-Ingersoll-Ross model. This relies on a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach a neighbourhood of zero. The method is hybrid in the sense that a convergent backstop method is invoked if the timestep becomes too small, or to prevent solutions from overshooting zero and becoming negative. Under parameter constraints that imply Feller's condition, we prove that such a scheme is strongly convergent, of order at least 1/2. Control of the strong error is important for multi-level Monte Carlo techniques. Under Feller's condition we also prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. Numerically, we compare this adaptive method to fixed step implicit and explicit schemes, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive in a domain that extends beyond Feller's condition, indicating suitability for the modelling of stochastic volatility in Heston-type asset models.
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