# ON EXACT CONVERGENCE RATE OF STRONG NUMERICAL SCHEMES FOR STOCHASTIC DIFFERENTIAL EQUATIONS

• Nam, Dou-Gu (National Institute for mathematical Sciences)
• Published : 2007.02.28
• 73 4

#### Abstract

We propose a simple and intuitive method to derive the exact convergence rate of global $L_{2}-norm$ error for strong numerical approximation of stochastic differential equations the result of which has been reported by Hofmann and $M{\"u}ller-Gronbach\;(2004)$. We conclude that any strong numerical scheme of order ${\gamma}\;>\;1/2$ has the same optimal convergence rate for this error. The method clearly reveals the structure of global $L_{2}-norm$ error and is similarly applicable for evaluating the convergence rate of global uniform approximations.

#### Keywords

strong approximation of SDE;global $L_2$-norm error

#### References

1. S. Cambanis and Y. Hu, Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design, Stochastics Stochastic Rep. 59 (1996), no. 3-4, 211-240 https://doi.org/10.1080/17442509608834090
2. N. Hofmann, T. Muller-Gronbach, and K. Ritter, The optimal discretization of stochastic differential equations, J. Complexity 17 (2001), no. 1, 117-153 https://doi.org/10.1006/jcom.2000.0570
3. N. Hofmann, T. Muller-Gronbach, and K. Ritter, Linear vs. standard information for scalar stochastic differential equations, J. Complexity 18 (2002), no. 2, 394-414 https://doi.org/10.1006/jcom.2001.0627
4. N. Hofmann and T. Muller-Gronbach, On the global error of It^o-Taylor schemes for strong approximation of scalar stochastic differential equations, J. Complexity 20 (2004), no. 5, 732-752 https://doi.org/10.1016/j.jco.2003.09.004
5. T. Muller-Gronbach, The optimal uniform approximation of systems of Stochastic differential equations, Ann. Appl. Probab. 12 (2002), no. 2, 664-690 https://doi.org/10.1214/aoap/1026915620
6. N. J. Newton, An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times. Stochastics Stochastic Rep. 29 (1990), no. 2, 227-258 https://doi.org/10.1080/17442509008833616
7. N. Hofmann, T. Muller-Gronbach, and K. Ritter, Optimal approximation of stochastic differential equations by adaptive step-size control, Math. Comp. 69 (2000), no. 231, 1017-1034 https://doi.org/10.1090/S0025-5718-99-01177-1
8. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992