Hybrid, adaptive, and positivity preserving numerical methods for the Cox-Ingersoll-Ross model
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We introduce an adaptive Euler method for the approximate solution of the Cox-Ingersoll-Ross short rate model. An explicit discretisation is applied over an adaptive mesh to the stochastic differential equation (SDE) governing the square root of the solution, relying upon 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 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. 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 schemes extant in the literature, both implicit and explicit, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive over the entire domain of Feller's condition.
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