论文标题
适应性在Cox-Ingersoll-Ross模型的数值方法中的作用
The role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model
论文作者
论文摘要
我们证明了自适应显式Euler方法在Cox-Ingersoll-Ross模型的近似解中的有效性。这依赖于一类路径的时间播放策略，这些策略通过在解决方案接近零附近时减少步骤来起作用。该方法是混合的，因为如果时间步有太小，或者防止解决方案超过零并变为负面，则会调用收敛的后退方法。在参数约束下，这意味着Feller的状况，我们证明了这种方案是强烈收敛的，至少为1/2。控制强误差对于多级蒙特卡洛技术很重要。在Feller的条件下，我们还证明，需要对预防负值进行后退方法的概率可以任意规模。从数值上讲，我们将这种自适应方法与固定步骤隐式和显式方案以及一种新颖的半平均自适应变体进行了比较。我们观察到，自适应方法会导致在范围内具有竞争力的方法，该域延伸超出了Feller的状况，这表明对Heston型资产模型中随机波动的建模适用。
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.
