In all previous work on Lyapunov exponents for stochastic systems, the results take the form of asymptotic power series expansions in the noise parameter. In this paper we consider a natural class of systems driven by white noise for which the asymptotic expansions containfractional powersof the noise parameter. In particular we obtain anexact formulafor certain cases. The systems studied include the free particle with multiplicative white noise and mechanical systems with one degree of freedom. The results are stated and proved for small noise and large noise