Semi-implicit Euler-Maruyama approximation for noncolliding particle systems. (English) Zbl 1464.60083

Ngo, Hoang-Long; Taguchi, Dai

Ann. Appl. Probab. 30, No. 2, 673-705 (2020).

Summary: We introduce a semi-implicit Euler-Maruyama approximation which preserves the noncolliding property for some class of noncolliding particle systems such as Dyson-Brownian motions, Dyson-Ornstein-Uhlenbeck processes and Brownian particle systems with nearest neighbor repulsion, and study its rates of convergence in both \(L^p\)-norm and pathwise sense.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)

Keywords:

implicit Euler-Maruyama approximation; noncolliding particle systems; rate of convergence

Software:

RODAS

References:

[1]	Alfonsi, A. (2013). Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process. Statist. Probab. Lett. 83 602-607. · Zbl 1273.65007 · doi:10.1016/j.spl.2012.10.034
[2]	Bru, M.-F. (1989). Diffusions of perturbed principal component analysis. J. Multivariate Anal. 29 127-136. · Zbl 0687.62048 · doi:10.1016/0047-259X(89)90080-8
[3]	Bru, M.-F. (1991). Wishart processes. J. Theoret. Probab. 4 725-751. · Zbl 0737.60067 · doi:10.1007/BF01259552
[4]	Cépa, E. and Lépingle, D. (1997). Diffusing particles with electrostatic repulsion. Probab. Theory Related Fields 107 429-449. · Zbl 0883.60089
[5]	Dereich, S., Neuenkirch, A. and Szpruch, L. (2012). An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 1105-1115. · Zbl 1364.65013 · doi:10.1098/rspa.2011.0505
[6]	Dyson, F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191-1198. · Zbl 0111.32703 · doi:10.1063/1.1703862
[7]	Graczyk, P. and Malecki, J. (2014). Strong solutions of non-colliding particle systems. Electron. J. Probab. 19 no. 119, 21. · Zbl 1307.60079 · doi:10.1214/EJP.v19-3842
[8]	Gyöngy, I. (1998). A note on Euler’s approximations. Potential Anal. 8 205-216. · Zbl 0946.60059
[9]	Hairer, E. and Wanner, G. (1991). Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics 14. Springer, Berlin. · Zbl 0729.65051
[10]	Hefter, M. and Jentzen, A. (2019). On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes. Finance Stoch. 23 139-172. · Zbl 1425.91401 · doi:10.1007/s00780-018-0375-5
[11]	Hutzenthaler, M., Jentzen, A. and Noll, M. (2014). Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries. Preprint. Available at arXiv:1403.6385.
[12]	Inukai, K. (2006). Collision or non-collision problem for interacting Brownian particles. Proc. Japan Acad. Ser. A Math. Sci. 82 66-70. · Zbl 1107.60041 · doi:10.3792/pjaa.82.66
[13]	Jentzen, A., Kloeden, P. E. and Neuenkirch, A. (2009). Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. Numer. Math. 112 41-64. · Zbl 1163.65003 · doi:10.1007/s00211-008-0200-8
[14]	Katori, M. and Tanemura, H. (2004). Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45 3058-3085. · Zbl 1071.82045 · doi:10.1063/1.1765215
[15]	Lépingle, D. (2010). Boundary behavior of a constrained Brownian motion between reflecting-repellent walls. Probab. Math. Statist. 30 273-287. · Zbl 1225.60060
[16]	Li, X. H. and Menon, G. (2013). Numerical solution of Dyson Brownian motion and a sampling scheme for invariant matrix ensembles. J. Stat. Phys. 153 801-812.
[17]	Mitrinovic, D. S., Pecaric, J. E. and Fink, A. M. (1994). Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications (East European Series) 53. Kluwer Academic, Dordrecht.
[18]	Neuenkirch, A. and Szpruch, L. (2014). First order strong approximations of scalar SDEs defined in a domain. Numer. Math. 128 103-136. · Zbl 1306.60075 · doi:10.1007/s00211-014-0606-4
[19]	Pettersson, R. (1995). Approximations for stochastic differential equations with reflecting convex boundaries. Stochastic Process. Appl. 59 295-308. · Zbl 0841.60042 · doi:10.1016/0304-4149(95)00040-E
[20]	Ramanan, K. and Shkolnikov, M. (2018). Intertwinings of beta-Dyson Brownian motions of different dimensions. Ann. Inst. Henri Poincaré Probab. Stat. 54 1152-1163. · Zbl 1391.60145 · doi:10.1214/17-AIHP835
[21]	Rogers, L. C. G. and Shi, Z. (1993). Interacting Brownian particles and the Wigner law. Probab. Theory Related Fields 95 555-570. · Zbl 0794.60100 · doi:10.1007/BF01196734
[22]	Rost, H. and Vares, M. E. (1985). Hydrodynamics of a one-dimensional nearest neighbor model. In Particle Systems, Random Media and Large Deviations (Brunswick, Maine, 1984). Contemp. Math. 41 329-342. Amer. Math. Soc., Providence, RI. · Zbl 0572.60095
[23]	Slominski, L. (1994). On approximation of solutions of multidimensional SDEs with reflecting boundary conditions. Stochastic Process. Appl. 50 197-219. · Zbl 0799.60055 · doi:10.1016/0304-4149(94)90118-X
[24]	Slominski, L. · Zbl 1262.60081 · doi:10.1016/j.spa.2012.10.006