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I will study from the probability thory to its application to stochastic differential equations with my friend. Of cource I'm looking forward to study them but would be a littel discouraging because I don't know their application to practical problems. To say that, I know the fact that they are practicaly used in companies etc. It is that I want to know what kind of scene they are really used in. Although one might say that SDEs or SPDEs are more natural than usual DEs or PDEs, I think that there is no difference between SDEs (SPDEs) and DEs (PDEs) in practical scenes since they are only mathematical subjects. (← This is only my own think!)

I'm glad if you tell me the practicality of SDEs and SPDEs.

And also, if I study while classifying practical application into the field of vision, should I begin study from which of SDEs or SPDEs after understanding basical tools?

Please give me comments.

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    $\begingroup$ Do you know whats the difference between deterministic and stochastic behavior in the real world? A single example would be you could try to model trends with deterministic equation such as the temperature in a room with a simple ode such as Newtons cooling, but you could look at fluctuations and try to model that with some form of stochastic process especially if you have sensor error (Though it is not something I have done personally). If you want to look at an area of mathematics that SDES are common, it would be Financial mathematics in the branch of Option/Derivative pricing. $\endgroup$
    – Chinny84
    Commented Nov 9, 2015 at 15:37
  • $\begingroup$ @Chinny84 Are'nt SDEs deterministic? I didn't know... It may be so as it were said. So the value of solutions for SDEs means a value under a certain probability? As you gave, I understand that SDEs can be used if we consider phenomena under some error, but I think also that it is anxiety in practical scene not to be able to say categolicaly that you should do it this way in this situation. $\endgroup$
    – user
    Commented Nov 9, 2015 at 16:01
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    $\begingroup$ mathematical finance, mathematical economics, financial engineering, physics, biology -- anywhere that has randomness $\endgroup$
    – BCLC
    Commented Nov 21, 2015 at 16:18

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Generally, the main difference with pdes is adding a degenerate process as a potential term that simulates random forcing from the environment. In statistics-ling it represents the sum total of all the confound variables that we can't identify but still affect the system.

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