  Stochastic differential equations (SDE) often exhibit large random
transitions. This property, which we denote as pathwise stiffness, causes
transient bursts of stiffness which limit the allowed step size for common
fixed time step explicit and drift-implicit integrators. We present four
separate methods to efficiently handle this stiffness. First, we utilize a
computational technique to derive stability-optimized adaptive methods of
strong order 1.5 for SDEs. The resulting explicit methods are shown to exhibit
substantially enlarged stability regions which allows for them to solve
pathwise stiff biological models orders of magnitude more efficiently than
previous methods like SRIW1 and Euler-Maruyama. Secondly, these integrators
include a stiffness estimator which allows for automatically switching between
implicit and explicit schemes based on the current stiffness. In addition,
adaptive L-stable strong order 1.5 implicit integrators for SDEs and stochastic
differential algebraic equations (SDAEs) in mass-matrix form with additive
noise are derived and are demonstrated as more efficient than the explicit
methods on stiff chemical reaction networks by nearly 8x. Lastly, we developed
an adaptive implicit-explicit (IMEX) integration method based off of a common
method for diffusion-reaction-convection PDEs and show numerically that it can
achieve strong order 1.5. These methods are benchmarked on a range of problems
varying from non-stiff to extreme pathwise stiff and demonstrate speedups
between 5x-6000x while showing computationally infeasibility of fixed time step
integrators on many of these test equations.
