Analysis of asymptotic mean-square stability of a class of Runge–Kutta schemes for linear systems of stochastic differential equations

2014 ◽  
Vol 105 ◽  
pp. 17-48 ◽  
Author(s):  
A. Haghighi ◽  
S.M. Hosseini
2003 ◽  
Vol 6 ◽  
pp. 297-313 ◽  
Author(s):  
Desmond J. Higham ◽  
Xuerong Mao ◽  
Andrew M. Stuart

AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.


2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Ying Du ◽  
Changlin Mei

Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-stepθmethod is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability.


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