The Stochastic Logarithmic Norm for Stability Analysis of Stochastic Differential Equations
It extends a classical tool from ODEs to SDEs, providing a new method for stability analysis of stochastic systems.
The paper introduces the stochastic logarithmic norm to analyze the stability of Itô stochastic differential equations with multiplicative noise, deriving stability estimates for linear SDEs in various norms and illustrating with examples.
To analyze the stability of Itô stochastic differential equations with multiplicative noise, we introduce the stochastic logarithmic norm. The logarithmic norm was originally introduced by G. Dahlquist in 1958 as a tool to study the growth of solutions to ordinary differential equations and for estimating the error growth in discretization methods for their approximate solutions. We extend the concept to the stability analysis of Itô stochastic differential equations with multiplicative noise. Stability estimates for linear Itô SDEs using the one, two and $\infty$-norms in the $l$-th mean, where $1 \leq l < \infty $, are derived and the application of the stochastic logarithmic norm is illustrated with examples.