Abstract
For the CIR equation \( {\text{d}}{X_t} = \left( {\theta - k\,{X_t}} \right){\text{d}}t + \sigma \sqrt {{{X_t}}} {\text{d}}{B_t} \), we propose positive weak first- and second-order approximations that use, at each step, generation of discrete (respectively two- and three-valued) random variables (Theorems 3 and 4). The equation is split into deterministic part \( {\text{d}}{D_t} = \left( {\theta - k{D_t}} \right){\text{d}}t \), which is solved exactly, and stochastic part \( {\text{d}}{S_t} = \sigma \sqrt {{{S_t}}} {\text{d}}{B_t} \), which is actually approximated in distribution.
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Mackevičius, V. Weak approximation of CIR equation by discrete random variables. Lith Math J 51, 385–401 (2011). https://doi.org/10.1007/s10986-011-9134-4
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DOI: https://doi.org/10.1007/s10986-011-9134-4