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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

Nikolaos Halidias 1,

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.
Keywords: positivity preserving, multidimensional super linear stochastic differential equations, Explicit numerical scheme, stochastic Predator-Prey model..
Mathematics Subject Classification: Primary: 60H10, 60H3.
Citation: Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete and 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), 79-87. doi: 10.1007/s11075-013-9724-9.

[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), 2083-2100. doi: 10.3934/dcdsb.2013.18.2083.

[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), 1041-1063. doi: 10.1137/S0036142901389530.

[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), 1611-1641. doi: 10.1214/11-AAP803.

[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., (). 

[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), 389-400. doi: 10.1016/j.amc.2013.08.023.

[7]

A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs with values in a domain, Num. Math., 128 (2014), 103-136. doi: 10.1007/s00211-014-0606-4.

show all references

References:
[1]

N. Halidias, A novel approach to construct numerical methods for stochastic differential equations, Numer Algor, 66 (2014), 79-87. doi: 10.1007/s11075-013-9724-9.

[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), 2083-2100. doi: 10.3934/dcdsb.2013.18.2083.

[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), 1041-1063. doi: 10.1137/S0036142901389530.

[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), 1611-1641. doi: 10.1214/11-AAP803.

[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., (). 

[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), 389-400. doi: 10.1016/j.amc.2013.08.023.

[7]

A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs with values in a domain, Num. Math., 128 (2014), 103-136. doi: 10.1007/s00211-014-0606-4.

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