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Thread: Stochastic Differential and Fokker-Planck equations

  1. #1
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    Stochastic Differential and Fokker-Planck equations

    Hello everyone,

    I am faced with a problem of having two coupled stochastic differential equations which needs to be solved through applying methods of solution w.r.t to corresponding Fokker-Planks equation, the two equations which is in question are given as:

    \frac{\partial x_{1}(X,t)}{\partial t} = -\frac{1}{2}\left(x_{1}(X,t) -x_{2}(X,t)\right) - \frac{1}{2}\xi(t)(x_{1}(X,t)-x_{2}(X,t))

    and


    \frac{\partial x_{2}(X,t)}{\partial t} = -\frac{1}{2}\left(x_{2}(X,t) -x_{1}(X,t)\right) - \frac{1}{2}\xi(t)(x_{2}(X,t)-x_{1}(X,t))

    Where \xi(t) is a gaussian noise process. For which I obtain the following Fokker-Planck equation:


    \frac{\partial P(X,t)}{\partial t} = - \sum \limits_{i=1}^{2} \frac{\partial }{\partial x_{i}}A_{i}(x)P(X,t) + \frac{1}{2} \sum \limits_{i,j=1}^{2} \frac{\partial^2 }{\partial x_{i}x_{j}}B_{i,j}(x)P(X,t)

    With A_{i} = \begin{pmatrix} -\frac{1}{2}x_{1}+ \frac{1}{2}x_{2} \\  -\frac{1}{2}x_{2} + \frac{1}{2}x_{1}\end{pmatrix}
    and B_{i,j} = \begin{pmatrix} - \frac{1}{2}(x_{1}(X,t)-x_{2}(X,t)) & 0 \\ 0 & - \frac{1}{2}(x_{2}(X,t)-x_{1}(X,t)) \end{pmatrix}

    Although this is the point in which I get stuck, I have no idea from where I need to proceed now, can anyone give a hint as to how I can't any further finding a solution?

    Kind Regards
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  2. #2
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    Re: Stochastic Differential and Fokker-Planck equations

    Hey Krisly.

    Do you need an analytic solution or does a numerical solution suffice?
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  3. #3
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    Re: Stochastic Differential and Fokker-Planck equations

    Hey Chiro

    Preferably an analytic solution. I have tried with eigenfunction expansion however I need to find the stationary solution, if it exists, which is obtained by solving:

    - \sum \limits_{i=1}^{2} \frac{\partial}{\partial x_i}A_i(x)P(X,t) + \frac{1}{2} \sum \limits_{i,j=1}^{2}\frac{\partial^2}{\partial x_i x_j}B_{i,j}P(X,t)=0

    However this doesn't seem any easier to solve.
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  4. #4
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    Re: Stochastic Differential and Fokker-Planck equations

    I know this may not give a solution per se, but you could try modelling the expectation function with Girsanov and seeing if you get something similar to another distribution you know.

    At least that can give you an idea of where the stochastic process is drifting to and later you can work out the details with regards to the residuals.
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