Multilevel Monte Carlo methods for positivity-preserving approximations of the Heston 3/2-model
<|reference_start|>Multilevel Monte Carlo methods for positivity-preserving approximations of the Heston 3/2-model: This article is concerned with the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model from mathematical finance, which takes values in $(0, \infty)$ and possesses superlinearly growing drift and diffusion coefficients. To discretize the SDE model, a new Milstein-type scheme is proposed to produce independent sample paths. The proposed scheme can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size $h>0$. This positivity-preserving property for large discretization time steps is particularly desirable in the MLMC setting. Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is not trivial, as the diffusion coefficient grows super-linearly. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity $\mathcal{O}(\epsilon^{-2})$ for the MLMC approach, where $\epsilon > 0$ is the required target accuracy. Numerical experiments are finally reported to confirm the theoretical findings.<|reference_end|>
@article{wu2024multilevel,
    title={Multilevel Monte Carlo methods for positivity-preserving approximations
  of the Heston 3/2-model},
    author={Xiaojuan Wu, Siqing Gan},
    journal={arXiv preprint arXiv:2403.05837},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.05837},
    primaryClass={math.NA cs.NA}
}