Langevin Stochastic Differential Equation
Here, two different approaches, viz. FEMM and FMM, are presented for solving the uncertain Langevin SDE. The uncertainties are taken in their initial conditions as well as associated parameters in terms of TFN. The limit method for fuzzy arithmetic has been used as a tool to handle the FSDE.
Solution of Fuzzy Stochastic Differential Equations
As mentioned earlier that the uncertainties are taken in initial conditions as well as associated parameters, the problems may be classified into the following three cases.
For the first case, we have considered the initial conditions of the SDE as uncertain, viz. fuzzy. As such, we consider Equation 11.8 with initial condition as fuzzy and then we have
Now, if we apply this fuzzy concept for the EMM, then Equation 11.13 may be represented in the following way:
Furthermore, it is noticed that due to the fuzziness of the initial value, we get a series of approximated fuzzy solutions and these are
When the MM is used to handle this situation (Equation 11.13), then the approximation scheme may be represented in the following manner:
In this case, we assume the involve parameters (or the coefficients) only as fuzzy, then the modified form of Equation 11.8 may be written as
By applying the fuzzy concept for the EMM, Equation 11.16 may be represented as where
For the MM, the approximation scheme for Equation 11.16 may be written as
It may be pointed out that as the parameters are taken as fuzzy, we get a series of approxi-
Wi+1 (a)- Wi+1 (a)
mated fuzzy solutions in the а-cut form and these are w i+1 (a) = wi+1 (a)+ - .
Finally, both the initial condition and involved parameters are considered as fuzzy and then Equation 11.8 may be represented as
Applying the same fuzzy concept for the EMM, Equation 11.19 may now be written as
and for the MM, the approximation scheme becomes