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Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. (English) Zbl 1163.65003
Summary: We study the approximation of stochastic differential equations on domains. For this, we introduce modified Itô-Taylor schemes, which preserve approximately the boundary domain of the equation under consideration. Assuming the existence of a unique non-exploding solution, we show that the modified Itô-Taylor scheme of order \(\gamma \) has pathwise convergence order \(\gamma - \varepsilon \) for arbitrary \(\varepsilon > 0\) as long as the coefficients of the equation are sufficiently differentiable. In particular, no global Lipschitz conditions for the coefficients and their derivatives are required. This applies for example to the so called square root diffusions.

MSC:
65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
65L20 Stability and convergence of numerical methods for ordinary differential equations
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[1] Alfonsi A.: On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11, 355–384 (2005) · Zbl 1100.65007 · doi:10.1515/156939605777438569
[2] Allen E.: Modelling with Itô Stochastic Differential Equations. Springer, Dordrecht (2007) · Zbl 1130.60064
[3] Arnold L.: Random Dynamical Systems. Springer, Berlin (1998) · Zbl 0906.34001
[4] Bossy, M., Diop, A.: An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form |x| \(\alpha\) , \(\alpha\) [1/2,1). Working paper (2004)
[5] Deelstra G., Delbaen F.: Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stoch. Models Data Anal. 14, 77–84 (1998) · Zbl 0915.60064 · doi:10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2
[6] Fleury G.: Convergence of schemes for stochastic differential equations. Prob. Eng. Mech. 21, 35–43 (2005) · doi:10.1016/j.probengmech.2005.07.001
[7] Gaines J.G., Lyons T.J.: Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57, 1455–1484 (1997) · Zbl 0888.60046 · doi:10.1137/S0036139995286515
[8] Grüne L., Kloeden P.E.: Pathwise approximation of random ordinary differential equations. BIT 41, 711–721 (2001) · Zbl 0998.65010 · doi:10.1023/A:1021995918864
[9] Gyöngy I.: A note on Euler’s approximations. Potential Anal. 8, 205–216 (1998) · Zbl 0946.60059 · doi:10.1023/A:1008605221617
[10] Higham D.J., Mao X.: Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comp. Finance 8, 35–61 (2005)
[11] Higham D.J., Mao X., Stuart A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063 (2002) · Zbl 1026.65003 · doi:10.1137/S0036142901389530
[12] Hu, Y.: Semi-implicit Euler–Maruyama scheme for stiff stochastic equations. In: Koerezlioglu, H. (ed.) Stochastic Analysis and Related Topics V: The Silvri Workshop, Progr. Prob. 38, Boston, pp. 183–202 (1996) · Zbl 0848.60057
[13] Kahl C., Günther M., Rossberg T.: Structure preserving stochastic integration schemes in interest rate derivative modeling. Appl. Numer. Math. 58, 284–295 (2008) · Zbl 1141.65323 · doi:10.1016/j.apnum.2006.11.013
[14] Karlin S., Taylor H.M.: A Second Course in Stochastic Processes. Academic Press, New York (1981) · Zbl 0469.60001
[15] Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991) · Zbl 0734.60060
[16] Kloeden P.E., Neuenkirch A.: The pathwise convergence of approximation schemes for stochastic differential equations. LMS JCM 10, 235–253 (2007) · Zbl 1223.60051
[17] Kloeden P.E., Platen E.: Numerical Solution of Stochastic Differential equations, 3rd edn. Springer, Berlin (1999) · Zbl 0752.60043
[18] Lamba H., Mattingly J.C., Stuart A.M.: An adaptive Euler–Maruyama scheme for SDEs: convergence and stability. IMA J. Numer. Anal. 27, 479–506 (2007) · Zbl 1127.65005 · doi:10.1093/imanum/drl032
[19] Lord, R., Koekkoek, R., van Dijk, D.J.C.: A Comparison of Biased Simulation Schemes for Stochastic Volatility Models. Working Paper (2008) · Zbl 1198.91240
[20] Mao X., Marion G., Renshaw E.: Environmental Brownian noise suppresses explosions in populations dynamics. Stoch. Process. Apl. 97, 95–110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[21] Mao X.: Stochastic Differential Equations and their Applications. Horwood Publishing, Chichester (1997) · Zbl 0892.60057
[22] Milstein G.N., Tretjakov M.V.: Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients. SIAM J. Numer. Anal. 43, 1139–1154 (2005) · Zbl 1102.60059 · doi:10.1137/040612026
[23] Ninomiya S., Victoir N.: Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15, 107–121 (2008) · Zbl 1134.91524 · doi:10.1080/13504860701413958
[24] Talay D.: Résolution trajectorielle et analyse numérique des équations différentielles stochastiques. Stochastics 9, 275–306 (1983) · Zbl 0512.60041
[25] Yan L.: The Euler scheme with irregular coefficients. Ann. Probab. 30, 1172–1194 (2002) · Zbl 1020.60054 · doi:10.1214/aop/1029867124
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