scholarly journals The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Chunmei Shi ◽  
Yu Xiao ◽  
Chiping Zhang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Euler Method ◽  
Exponential Euler Method

scholarly journals On the Numerical Solution of Some Non-Linear Stochastic Differential Equations Using the Semi-Discrete Method

2016 ◽  
Vol 16 (1) ◽  
pp. 105-132 ◽  
Author(s):  
Nikolaos Halidias ◽  
Ioannis S. Stamatiou
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Stochastic Differential Equations ◽  
Numerical Experiments ◽  
Euler Method ◽  
Optimal Order ◽  
Financial Mathematics ◽  
Negative Solution ◽  
Discrete Method ◽  
Linear Problems

AbstractWe are interested in the numerical solution of stochastic differential equations with non-negative solutions. Our goal is to construct explicit numerical schemes that preserve positivity, even for super-linear stochastic differential equations. It is well known that the usual Euler scheme diverges on super-linear problems and the tamed Euler method does not preserve positivity. In that direction, we use the semi-discrete method that the first author has proposed in two previous papers. We propose a new numerical scheme for a class of stochastic differential equations which are super-linear with non-negative solution. The Heston 3/2-model appearing in financial mathematics belongs to this class of stochastic differential equations. For this model we prove, through numerical experiments, the “optimal” order of strong convergence at least 1/2 of the semi-discrete method.

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