In this article,we study for the first time the stochastic Kaup–Newell equation in birefringent fibers. Three integration algorithms are applied, namely, the unified Riccati equation expansion method, the addendum to Kudryashov's method, and the addendum to modified sub-ODE method. Many types of optical soliton solutions such as Jacobi-elliptic function solutions,Weierstrass elliptic function solutions, bright solitons, dark solitons, singular solitons, and periodic function solutions are obtained. Numerical schemes give a visual perspective to the solutions derived analytically.