NUMERICAL ANALYSIS OF REFLECTING BROWNIAN MOTION AND A NEW MODEL OF SEMI-REFLECTING BROWNIAN MOTION WITH SOME DOMAINS

[1] O. Faure, Numerical pathwise approximation of stochastic differential equations, Prepublication of LAMM , Ecole Nationale des Ponts et Chaussees, Paris, 1990.
[2] S. Kanagawa, On the rate of convergence for Maruyama’s approximate solutions of stochastic differential equations, Yokohama Mathematical Journal, 36: 79–86, 1988.
[3] S. Kanagawa, The rate of convergence for approximate solutions of stochastic differential equations, Tokyo J. Math., 12: 33–48, 1989.
[4] S. Kanagawa, Confidence intervals of discretized Euler-Maruyama approximate solutions of SDE’s, Nonlinear Analysis, Theory, Methods and Applications, 30: 4101–4104, 1997.
[5] S. Kanagawa, Some Remarks on Strong Approximation of Reflecting Brownian Motion Using Penalty Method, Proceedings of Neural, Parallel and Scientific Computations, 2: 63–70, 2002, Atlanta, U.S.A.
[6] S. Kanagawa, A. Arimoto and Y. Saisho, Numerical simulation of reflecting geometrical Brownian motion and its application to mathematical finance, Nonlinear Analysis, 63: e2209–e2222, 2005.

[7] S. Kanagawa, K. Inoue, A. Arimoto and Y. Saisho, Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method, Discrete and Continuous Dynamical Systems, Supplement Volume: 463–475, 2005, American Institute of Mathematical Sciences.
[8] S. Kanagawa and S. Ogawa, Numerical Solutions of Stochastic Differential Equations and Their Applications, Sugaku Expositions, Amer. Math. Soc., 18 : 75–99, 2005.
[9] S. Kanagawa and Y. Saisho, Strong Approximation of Reflecting Brownian Motion Using Penalty Method and its Application to Computer Simulation, Monte Carlo Methods Appl., 6: 105-114, 2000.
[10] S. Kanagawa, Y. Saisho and H. Uesu, Error Estimation for the Euler-Maruyama Approximate Solution of Reflecting Brownian Motion Using Penalty Method and Numerical Applications, Dynamic Systems and Applications, 15: 287–300, 2006.
[11] P.L. Lions and A.S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37: 511–537, 1981.
[12] D. L ́epingle, Euler scheme for reflected stochastic differential equations, Proceedings of Probabilit ́es numeriqu ́es, Paris 1992, in Math. Comput. Simulation, 38: 119–126, 1995.
[13] G. Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo, 4: 48–90, 1955.
[14] R. Pettersson, Approximation for stochastic differential equations with reflecting convex boundaries, Stoch. Processes Appl., 59: 295–308, 1995.
[15] Y. Saisho, Stochastic differential equations for multi-dimensional domain with reflecting boundary, Probab. Theory Rel. Fields, 74: 455–477, 1987.
[16] Y. Saisho and H. Tanaka, Stochastic differential equations for mutually reflecting Brownian balls, Osaka J. Math., 23: 725–740, 1986.
[17] Y. Saisho and H. Tanaka, On the symmetry of a reflecting Brownian motion defined by Skorohod’s equation for a multi-dimensional domain, Tokyo J.Math., 10: 419–435, 1987.
[18] H. Schurz, Numerical Analysis of SDEs without tears, in V. Lakshmikantham and D. Kannan (eds.) Handbook of Stochastic Analysis and Applications, Mecel Dekker, Basel, 237–359, 2002.
[19] H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J., 9: 163-177, 1979.
[20] K. Yoshihara, Weakly Dependent Stochastic Sequences and Their Applications, 4, Curve Estimation based on Weakly Dependent Data, Sanseido Co. Ltd., Tokyo, 1994.