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Deriving Probability Density Functions from Partial Differential Equations?

  1. Jun 26, 2012 #1
    Deriving Probability Density Functions from Partial Differential Equations???

    Hiyas,

    I have been told that it is quite normal to get PDFs (Probability Density Functions) from PDEs (Partial Differential Equations). That in PDEs that the function can be a PDF and you can get this by solving the PDE.

    Nobody has actually shown me how this is done. I have my doubts because I have learned that Probability Theory is not mixed with PDEs directly this way. That there are other types of mathematics called Stochastic Partial Differential Equations or Stochastic Differential Equations that do something similar.

    I was taught to believe that PDFs are found by "reinterpreting" the PDE by approaching it with Probability Theory and looking at from a new way and creating new Probability Mathematics to deal with the PDE.

    However others are insisting that I don't start by approaching it with Probability Theory. That I actually just solve PDEs to get PDFs and that I can also get PDFs from PDEs.

    Another thing is they are using the term Probability Distribution Function and not Probability Density Function but I think they are claiming them to be the same thing.

    Can anyone give me their thoughts on this.

    Thank you.
     
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  3. Jun 26, 2012 #2

    chiro

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    Re: Deriving Probability Density Functions from Partial Differential Equations???

    Hey Annihilator and welcome to the forums.

    The common area for this in financial calculus. What happens is you reduce a stochastic differential equation to a PDE and then you solve it.

    One good thing about the PDE approach is that when you get arbitrary pay-off functions for your option contract, you can get an expression for the expectation of that arbitrary function.

    The heat equation and the normal distribution are pretty much the same, so when you have the right transformations, you can use things like the heat equation to get a density or expectation value.

    The above though is a specialized application of this.

    There is however a common tool in probability that is known as the characteristic function which is used to retrieve PDF's from a MGF (Moment Generating Function) where you do a fourier transform on the MGF.

    I'm speculating, but it's quite possible that this transform or others like it are able to be translated to PDE's or other similar representations especially if a representation is found to link the fourier transform with PDE's. I know you can relate these to linear ODE's but not sure about PDE's.
     
  4. Jun 26, 2012 #3

    Mute

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    Re: Deriving Probability Density Functions from Partial Differential Equations???

    A PDE tells you how a given function changes with respect to its variables (time, position, etc). If your PDF changes with time and position (i.e., the values you can measure for your random variable change with time and/or position), then one can in principle write down an equation which relates together how the probability changes with time and space, and then solve this equation for the pdf.

    See http://en.wikipedia.org/wiki/Master_equation and http://en.wikipedia.org/wiki/Fokker-Planck_equation .
     
  5. Jun 26, 2012 #4
    Re: Deriving Probability Density Functions from Partial Differential Equations???

    Deriving probability density functions from pde's is also quite common in classical physics like turbulence. You can go from the Navier stokes equations (pde) to the transport equation of the probability density function of velocity (a Fokker-Planck equation) using not much more than the ito formula and standard calculus. The solution of the Fokker-Planck equation can be obtained by realizing that a stochastic sample of the PDF (where the PDF = the solution of the FP equation) is actually described by a Langevin equation. A discrete solution of the FP equation can be obtained by solving N Langevin equations in a Monte Carlo method.

    Actually, every PDE can be written as a stochastic equation, either as a Fokker-Planck equation or a Langevin equation (because every (?) PDE can be written as a diffusion process).

    Check the book of Oksendal for an introduction.
     
  6. Jun 26, 2012 #5

    Mute

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    Re: Deriving Probability Density Functions from Partial Differential Equations???

    Be warned, that book is rather formal and makes extensive use of measure theory notation. If you're comfortable with formal proofs and measure theory, go for it, but coming from a physics background I found it pretty dense and hard to read through.
     
  7. Jun 27, 2012 #6
    Re: Deriving Probability Density Functions from Partial Differential Equations???

    Ok, nice thank you.

    I take it though that not all PDEs can be written as a type of diffusion process without introducing some form of normalization of the PDE so that you can then begin Stochastic calculations. It seems to me some of these master equations types are pure postulates that can't be derived by any general means of getting PDFs from PDEs. It seems more like you have to figure out if the PDE can be modelled as a SPDE or SDE first and then get the PDF from there. Is this right or wrong?
     
  8. Jun 27, 2012 #7

    Mute

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    Re: Deriving Probability Density Functions from Partial Differential Equations???

    It depends on what you want to do, I guess. If you want analytical results, I think you write down an SDE and then try to convert it into a PDE, because we know how to solve PDEs much better than we know how to solve SDEs.

    However, if you want numerical results, it's easer to have an SDE which you can simulate because then you just add some noise terms which are really easy to do numerically; easier than solving PDEs numerically, at least.

    See

    http://en.wikipedia.org/wiki/Feynman-Kac_formula

    That discusses the connection between parabolic PDEs and SDEs.
     
  9. Jun 27, 2012 #8
    Re: Deriving Probability Density Functions from Partial Differential Equations???

    Cool. I know we are thinking the same because I am looking at the very same thing.

    By the way have you heard of cases where people working on PDEs didn't know they where working Stochastic Partial Differential Equations or Stochastic Differential Equations and didn't get the differences at post-graduate level?

    Thanks.
     
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