The Stochastic Logarithmic Norm for Stability Analysis of Stochastic Differential Equations

To analyze the stability of It\^o 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 It\^o stochastic differential equations with multiplicative noise. Stability estimates for linear It\^o 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.