The goal of this study is to see if there is a solution for the fuzzy delay predator-prey system (FDPPS) with Caputo derivative. To begin, we use Schaefer's fixed point theorem to obtain results for the existence theorem of at least one solution in a Caputo FDPPS where the initial condition is also represented by a fuzzy number on fuzzy number space. We also determine the necessary and sufficient conditions of solutions for the system. Several examples are also presented to explain the oscillatory property and the existence of a solution.

In this manuscript, we formulate the system of fuzzy stochastic fractional evolution equations (FSFEEs) driven by fractional Brownian motion. We find the results about the existence-uniqueness of the formulated system by using the Lipschitizian conditions. By using these conditions we have also investigated the exponential stability of the solution for the above system driven by fractional Brownian motion. Finally, the applications in financial mathematics are presented and the use of financial mathematics in the fractional Black and Scholes model is also discussed. An example is propounded to show the applicability of our results.