  We introduce a tamed exponential time integrator which exploits linear terms
in both the drift and diffusion for Stochastic Differential Equations (SDEs)
with a one sided globally Lipschitz drift term. Strong convergence of the
proposed scheme is proved, exploiting the boundedness of the geometric Brownian
motion (GBM) and we establish order 1 convergence for linear diffusion terms.
In our implementation we illustrate the efficiency of the proposed scheme
compared to existing fixed step methods and utilize it in an adaptive time
stepping scheme. Furthermore we extend the method to nonlinear diffusion terms
and show it remains competitive. The efficiency of these GBM based approaches
are illustrated by considering some well-known SDE models.
