This paper explores the dynamics of highly dispersive gap solitons within the framework of the Kundu-Eckhaus equation, augmented by the inclusion of multiplicative white noise in the Itô sense, a novel addition to the model. Two integration schemes, the extended simplest equation approach and the generalized Riccati equation mapping scheme, are employed to analyze and integrate the model. Despite yielding bright, singular, and dark-singular straddled solitons, both methods independently fail to capture dark optical solitons. Additionally, our investigation highlights that the influence of white noise predominantly affects the phase aspect of the solitons, with negligible impact on their amplitude. Further details regarding the specific physical system or optical medium under study would provide readers with context, aiding in the comprehension of the significance of our findings.