Hybridized Uncertainty in Point Kinetic Diffusion
As mentioned earlier, every system possesses uncertainties and they occur due to partial knowledge and truth. In general, partial knowledge-based uncertainties may be handled by probability theory and truth-based uncertainties are operated through possibility theory. In practical problems, we may have the combined effect of both the uncertainties. As such, this chapter comprises hybridization of the concept of stochasticity with fuzzy theory. Here, we have the modelled SDE, which includes the essence of fuzziness, and we call it the fuzzy stochastic differential equation (FSDE). Two well-known methods, viz. Euler-Maruyama method (EMM) and Milstein method (MM), are extended to the fuzzy form, and these are named fuzzy Euler-Maruyama method (FEMM) and fuzzy Milstein method (FMM). These methods have been used to investigate various diffusion problems.
Black-Scholes Stochastic Differential Equation
The Black-Scholes SDE with fuzzy uncertainty is investigated here. The uncertainties are assumed to occur due to the parameters involved in the system and these are considered as triangular fuzzy numbers (TFNs). The fuzzy arithmetic (Nayak and Chakraverty 2013) is used as a tool to handle FSDE. In particular, a system of Ito stochastic differential equations is analyzed with the fuzzy parameters. Furthermore, exact and EM approximation methods with fuzzy uncertainties are demonstrated.
For the sake of completeness, initially we discuss the crisp SDE and it is solved by using the Ito integral technique. Furthermore, the same problems are discussed for uncertain cases and the corresponding FSDEs are solved.