If you think it is for the best, please give an example where it made things easier or made a better model, and if possible some applications. Otherwise please give your reasons with example what went wrong, or where it made things more complex.
Best regards
Sarmad.
Dear Sarmad, I do not think that it is a natural merge of two subfields of mathematics. I do not know any practical problem calling for that (but I might be ignorant). It might also be technically difficult.
PDE are mostly used for spatio-temporal evolution of deterministic systems (Stochastic PDE is a minority). Fuzzy set is used to describe something that we do not know exactly. So mathematics of exact merging with non-exact looks a bit strange for me.
I know that PDE were successfully merged with topology and geometry, by consideration them on manifolds.
The fuzzy set theory initiated by L.A. Zadeh provided mathematicians with an
appropriate tool for modelling the vagueness phenomenon and shed new light into
the control theory for engineers. Later, in 1985, T. Takagi and M. Sugeno invented a
particular fuzzy model which became very popular due to its approximation ability.
Finally, in the 1990’s, several studies aimed at approximation properties of the other
widely used fuzzy models.
Regards!
Adriana.
Dear Adriana, I found in the text: "fuzzy approach to numerical solutions of differential equations" (p.69). This turns discussion into a different format. We are not trying to invent a new field of math (like fuzzy operator for PDE type equation - I have no idea how to derive it), but rather to elaborate a tool for numerical solution of PDE. Do I understand it correctly?
When we are trying to solve PDE, we start from exact solutions. For wave equations they also exist, but sometimes even for nonlinear PDE, like solitons of KdV. If we cannot solve exactly, there are different approximations. We can find a numerical solution by finite difference scheme, or to do an asymptotic expansion. You are talking just about one of numerical methods that can be useful in some cases. Maybe indeed in meteorology, when Lorenz instability makes the solution very sensitive to initial conditions, it makes sense to treat it as a "cloud", or fuzzy object.
Dear Yuri Yegorov;
Thank you for your answers, could you please tell me your opinion about " The application of nonlinear fuzzy parameters PDE method in pricing and hedging European options https://doi.org/10.1016/j.fss.2016.12.005 ", and this is one example of many.
Best regards
Sarmad
Dear Sarmad, I am a specialist in wave equations but not in option pricing models using fuzzy sets. I know that many physicists also contributed their ideas to finance. Finance swallows today all ideas. I just see that here coefficients are fuzzy, but not the derivative. Another possibility is to solve DE exactly and to get solution as a function of parameter, and then to use distribution function of parameter to see how the solution changes. This is what people would do in wave theory, but perhaps for finance fuzzy approach is better.
Yes, you convinced me that there are applications. Good luck!
Dear Sarmad the nonlinear fuzzy parameters PDE method in pricing and hedging European options is involving stochastic differential equations that transform the stochastic model into PDE then the estimate the model parameters by using one of the optimization methods the field in financial mathematics
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