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By using a change of scale and space, we study a class of stochastic differential equations (SDEs) whose solutions are drift-perturbed and exhibit asymptotic behaviour similar to standard Brow- nian motion. In particular sufficient conditions ensuring that these processes obey the Law of the Iterated Logarithm (LIL) are given. Ergodic-type theorems on the average growth of these non-stationary processes, which also depend on the asymptotic behaviour of the drift coefficient, are investigated. We apply these results to inefficient financial market models. The techniques extend to certain classes of finite-dimensional equation.

Solutions of Stochastic Differential Equations obeying the Law of the Iterated Logarithm, with applications to financial markets