This paper studies the asymptotic growth and decay properties of solutions of
the stochastic pantograph equation with multiplicative noise. We give
sufficient conditions on the parameters for solutions to grow at a polynomial
rate in p-th mean and in the almost sure sense. Under stronger conditions the
solutions decay to zero with a polynomial rate in p-th mean and in the almost
sure sense. When polynomial bounds cannot be achieved, we show for a different
set of parameters that exponential growth bounds of solutions in p-th mean
and an almost sure sense can be obtained. Analogous results are established for
pantograph equations with several delays, and for general finite dimensional
equations.Comment: 29 pages, to appear Electronic Journal of Qualitative Theory of
Differential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ.
(July 1--4, 2015, Szeged, Hungary
