T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise
The notion of T-stability concerns the behaviour of numerical methods (equipped with a driving process to be specified) when applied to the integration of stochastic ordinary differential equations. This concept can be easily extended, of course, to stochastic differential equations with delay. In essence, the numerical scheme is said to be T-stable if the solution it provides on a test linear equation tends to zero with probability one as time integration goes on. The author analyzes the T-stability of the semi-implicit Euler scheme applied to a linear delay differential equation with multiplicative noise (described by a standard Wiener process), giving conditions on the parameters of the method (the weight and step size) that guarantee this T-convergence. Now the Wiener process is approximated by a discrete random variable with two-point distribution (each with probability \(1/2\)). The numerical examples collected in the paper illustrate the obtained theoretical bounds for T-stability, and show that these are not optimal.
stochastic ordinary differential equations
scientific article; zbMATH DE number 5707080