Unconditionally positivity-preserving explicit Euler-type schemes for a generalized Ait-Sahalia model
CoRR(2024)
Abstract
The present work is devoted to strong approximations of a generalized
Ait-Sahalia model arising from mathematical finance. The numerical study of the
considered model faces essential difficulties caused by a drift that blows up
at the origin, highly nonlinear drift and diffusion coefficients and
positivity-preserving requirement. In this paper, a novel explicit Euler-type
scheme is proposed, which is easily implementable and able to preserve
positivity of the original model unconditionally, i.e., for any time step-size
h>0. A mean-square convergence rate of order 0.5 is also obtained for the
proposed scheme in both non-critical and general critical cases. Our work is
motivated by the need to justify the multi-level Monte Carlo (MLMC) simulations
for the underlying model, where the rate of mean-square convergence is required
and the preservation of positivity is desirable particularly for large
discretization time steps. To the best of our knowledge, this is the first
paper to propose an unconditionally positivity preserving explicit scheme with
order 1/2 of mean-square convergence for the model. Numerical experiments are
finally provided to confirm the theoretical findings.
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