## On the discretization schemes for the CIR (and Bessel squared) processes.(English)Zbl 1100.65007

The author considers the simulation of the Cox-Ingersoll-Ross (CIR) process defined by the stochastic integro-differential equation
$X_t=x_0+\int_0^t(a-kX_s)\, ds +\sigma\int_0^t\sqrt{X_s}\,dW_s,\quad t\in[0,T].$ He presents several discretization schemes of both implicit and explicit types, studies their strong and weak convergence, examines them numerically, and compares then with already known schemes.

### MSC:

 65C30 Numerical solutions to stochastic differential and integral equations 60H20 Stochastic integral equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 45R05 Random integral equations 65R20 Numerical methods for integral equations
Full Text:

### References:

 [1] DOI: 10.1007/BF01303802 · Zbl 0838.60051 [2] Bally V., Monte Carlo Methods Appl. 2 pp 93– (1996) [3] DOI: 10.1007/s00780-004-0131-x · Zbl 1065.60085 [4] DOI: 10.2307/1911242 · Zbl 1274.91447 [5] DOI: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2 · Zbl 0915.60064 [6] Diop, A. (2003). Sur la discretisation et le comportement a petit bruit d’EDS multidimensionnelles dont les coefficients sont a derivees singulieres, ph.D Thesis, INRIA. (available at http://www.inria.fr/rrrt/tu-0785.html) [7] Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering Springer, Series : Applications of Mathematics , Vol. 53. · Zbl 1038.91045 [8] Guyon, J. (2005). Euler scheme and tempered distributions, preprint CERMICS No. 277. [9] DOI: 10.1093/rfs/6.2.327 · Zbl 1384.35131 [10] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edition. Springer, Series : Graduate Texts in Mathematics, Vol. 113. · Zbl 0734.60060 [11] Lamberton, D. and Lapeyre, B. (1992). Une introduction au calcul stochastique applique a la finance, Ellipses. English version (1995): An Introduction to Stochastic Calculus Applied to Finance, Chapman and Hall. · Zbl 0898.60002 [12] DOI: 10.1137/S0036142901395588 · Zbl 1028.60064 [13] Rogers, L.C.G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales, 2nd edition. Cambridge Mathematical Library. · Zbl 0949.60003 [14] Talay D., Stochastic Analysis and Applications 8 (4) pp 94– (1990) · Zbl 0697.60066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.