Abstract

We introduce a concept of Lyapunov exponents and Lyapunov spectrum for nonautonomous linear stochastic differential equations. The Lyapunov exponents are defined samplewise via the two-parameter flow generated by the equation. We prove that Lyapunov exponents are finite and nonrandom. Lyapunov exponents are used for investigation of Lyapunov regularity and stability of nonautonomous stochastic differential equations. The results show that the concept of Lyapunov exponents is still very fruitful for stochastic objects and gives us a useful tool for investigating sample stability as well as qualitative behavior of nonautonomous linear and nonlinear stochastic differential equations.