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Stochastic differential equations (SDE) often exhibit large randomtransitions. This property, which we denote as pathwise stiffness, causestransient bursts of stiffness which limit the allowed step size for commonfixed time step explicit and drift-implicit integrators. We present fourseparate methods to efficiently handle this stiffness. First, we utilize acomputational technique to derive stability-optimized adaptive methods ofstrong order 1.5 for SDEs. The resulting explicit methods are shown to exhibitsubstantially enlarged stability regions which allows for them to solvepathwise stiff biological models orders of magnitude more efficiently thanprevious methods like SRIW1 and Euler-Maruyama. Secondly, these integratorsinclude a stiffness estimator which allows for automatically switching betweenimplicit and explicit schemes based on the current stiffness. In addition,adaptive L-stable strong order 1.5 implicit integrators for SDEs and stochasticdifferential algebraic equations (SDAEs) in mass-matrix form with additivenoise are derived and are demonstrated as more efficient than the explicitmethods on stiff chemical reaction networks by nearly 8x. Lastly, we developedan adaptive implicit-explicit (IMEX) integration method based off of a commonmethod for diffusion-reaction-convection PDEs and show numerically that it canachieve strong order 1.5. These methods are benchmarked on a range of problemsvarying from non-stiff to extreme pathwise stiff and demonstrate speedupsbetween 5x-6000x while showing computationally infeasibility of fixed time stepintegrators on many of these test equations.

Stability-Optimized High Order Methods and Stiffness Detection for Pathwise Stiff Stochastic Differential Equations