# American Institute of Mathematical Sciences

January  2015, 20(1): 153-160. doi: 10.3934/dcdsb.2015.20.153

## Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations

 1 Department of Mathematics, University of the Aegean, Karlovassi 83200 Samos, Greece

Received  October 2013 Revised  April 2014 Published  November 2014

In this note we work on the construction of positive preserving numerical schemes for a class of multidimensional stochastic differential equations. We use the semi discrete idea that we have proposed before proposing now a numerical scheme that preserves positivity on some multidimensional stochastic differential equations converging strongly in the mean square sense to the true solution.
Citation: Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153
##### References:
 [1] N. Halidias, A novel approach to construct numerical methods for stochastic differential equations,, Numer Algor, 66 (2014), 79. doi: 10.1007/s11075-013-9724-9. Google Scholar [2] D. J. Higham, X. Mao and L. Szpruch, Convergence, non-negativity and stability of a new Milstein scheme with applications to finance,, Discrete and Continuous Dynamical Systems - Series B, 18 (2013), 2083. doi: 10.3934/dcdsb.2013.18.2083. Google Scholar [3] D. Higham, X. Mao and A. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations,, SIAM J. Numer. Anal., 40 (2002), 1041. doi: 10.1137/S0036142901389530. Google Scholar [4] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients,, Ann. App. Probab., 22 (2012), 1611. doi: 10.1214/11-AAP803. Google Scholar [5] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients,, To appear in Mem. Amer. Math. Soc., (). Google Scholar [6] W. Liu and X. Mao, Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations,, Applied Mathematics and Computation, 223 (2013), 389. doi: 10.1016/j.amc.2013.08.023. Google Scholar [7] A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs with values in a domain,, Num. Math., 128 (2014), 103. doi: 10.1007/s00211-014-0606-4. Google Scholar

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##### References:
 [1] N. Halidias, A novel approach to construct numerical methods for stochastic differential equations,, Numer Algor, 66 (2014), 79. doi: 10.1007/s11075-013-9724-9. Google Scholar [2] D. J. Higham, X. Mao and L. Szpruch, Convergence, non-negativity and stability of a new Milstein scheme with applications to finance,, Discrete and Continuous Dynamical Systems - Series B, 18 (2013), 2083. doi: 10.3934/dcdsb.2013.18.2083. Google Scholar [3] D. Higham, X. Mao and A. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations,, SIAM J. Numer. Anal., 40 (2002), 1041. doi: 10.1137/S0036142901389530. Google Scholar [4] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients,, Ann. App. Probab., 22 (2012), 1611. doi: 10.1214/11-AAP803. Google Scholar [5] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients,, To appear in Mem. Amer. Math. Soc., (). Google Scholar [6] W. Liu and X. Mao, Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations,, Applied Mathematics and Computation, 223 (2013), 389. doi: 10.1016/j.amc.2013.08.023. Google Scholar [7] A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs with values in a domain,, Num. Math., 128 (2014), 103. doi: 10.1007/s00211-014-0606-4. Google Scholar
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