## Uniform approximation of the Cox-Ingersoll-Ross process.(English)Zbl 1335.65011

A simulation algorithm is developed for uniformly approximating the trajectories of the Cox-Ingersoll-Ross process $$V(t)$$ defined as the solution of the stochastic differential equation $dV(t)=k(\lambda-V(t))dt+\sigma\sqrt{V(t)}dw,\quad V(t_0)=V_0,$ where $$w$$ is a scalar Brownian motion and $$k, \lambda,\sigma$$ are positive constants. It is proved that the algorithm converges on any trajectory of $$V(t)$$ that is positive on $$[t_0,t_0+T]$$. A modification of the algorithm is devised to deal with trajectories that get close to zero, and its convergence is proved. Numerical implementation is briefly discussed with emphasis on examples from Finance.

### MSC:

 65C30 Numerical solutions to stochastic differential and integral equations 91G60 Numerical methods (including Monte Carlo methods) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 60J65 Brownian motion 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 33C15 Confluent hypergeometric functions, Whittaker functions, $${}_1F_1$$
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