when we need to solve the Fokker Planck equation (Kolmogorov Forward equation) with finite difference, we need to solve it in a bounded domain, (regardless of the dimension of the FPE), for more accurate solution, which kinds of boundary condition should be considered?
1-Natural boundary condition:
which is a Dirichlet type boundary condition
the value of probability at the boundaries equal to zero
2-the Reflecting boundary condition:
which I think is the Robin type boundary condition
and the Flux at boundaries is zero?
Dear Dr. Ali Namadchian
Greating!
The following links may be useful to answer
https://en.wikipedia.org/wiki/Kolmogorov_forward_equations
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http://www.eco.uc3m.es/~mkredler/ContTime/KolmForwEqu.pdf
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http://web.mit.edu/8.592/www/lectures/lec3/lec3.pdf
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https://cims.nyu.edu/~obuhler/Oliver_Buhler/Teaching_12_Spring_files/PDEfinance12notes1.pdf
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https://www.math.cmu.edu/~gautam/teaching/2011-12/880-advanced-scalc/pdfs/kolmogorov-forward.pdf
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https://pdfs.semanticscholar.org/2f51/93058a91c118a786e431204533a3d2c3e6d1.pdf
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https://www.ias.ac.in/article/fulltext/sadh/031/04/0445-0461
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https://www.sciencedirect.com/science/article/pii/S0022407307002683
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Chapter Numerical Solution of the Fokker–Planck Equation by Finite D...
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http://accelconf.web.cern.ch/accelconf/pac97/papers/pdf/4V039.PDF
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https://onlinelibrary.wiley.com/doi/pdf/10.1002/num.1690040305
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http://users.math.msu.edu/users/wei/paper/p24.pdf
With best wishes.
Dear,
See my published paper:
https://www.sciencedirect.com/science/article/pii/S2211379716300778
Dear Dr. Majeed Ahmed Weli
I could not find where you talk about boundary condition in your paper
I had implemented it with Dirichlet conditions on the bounds (0 density on the far bounds), which you correctly state as the natural solution. The more interesting bit is the initial condition: time-zero density is a Dirac mass at the starting value. One solution is carry the first time step analytically under a locally Gaussian approximation and start the FDM grid on time T1 with a Gaussian density boundary. I hope this helps.
Dear Dr. Antoine Savine
how was your results,
does the FPE solution remains positive for all times t?
what about conservation law?(at each time the integration of solution over the domain must remains equal to 1)
Hello Ali Namadchian and apologies for the late answer.
On the first time step T1, the weights on the node are positive and sum to 1 by construction. For there on, it is down to your FDM scheme.
1) I myself was targeting fast convergence so I applied a semi-explicit scheme that converges in (dt)^2 and (dx)^2 but does not guarantee positiveness or conservation of mass.
2) In order to guarantee that these two properties hold on every time step, you must:
a- Apply a fully implicit scheme, at the cost of a lower convergence order in (dt)
b- Work on edge conditions purposely, since standard Dirichlet will not give you the properties you want.
My friend Brian Norsk Huge did a lot of research in this field and published an award winning article with Jesper Andreasen (in a financial risk management context) where he shows that implementing the forward FP scheme as the adjoint of the backward scheme guarantees the properties you are seeking.
Kind regards,
Antoine
Brian Norsk Huge 's paper:
www.researchgate.net/publication/235622407_Random_grids
I hope this helps.
Dear Antoine Savine
Thank you very much for your answers.
I am trying to do as you say, using fully implicit method and work on boundary condition, also I find the Chang Cooper scheme useful
Article A practical difference scheme for Fokker-Planck equations
in this article reflecting boundary condition is used instead of natural boundary condition.
in your answer you mentioned:
" I myself was targeting fast convergence so I applied a semi-explicit scheme that converges in (dt)^2 and (dx)^2 but does not guarantee positiveness or conservation of mass"
may I ask you what was your goal when you solve FPE, because FPE gives the transient PDF of a stochastic process, So the PDF must be positive and sum to 1 over its domain.
I assumed you mean that although your solution does not satisfy these two condition, but the PDF you obtained can be used to derive the statistical characteristic of the system.
Brian Norsk Huge just posted a note that you may find helpful:
www.researchgate.net/publication/328580623_A_Note_on_the_Fokker-Planck_Equation
Kind regards,
Antoine
The best way to determine what BC apply to the Forward F-P equation you are solving is to keep in mind that the FFP equation solves for the probability diffusion and transport of a stochastic process as the process increases its time (or space) indexing. In other words, in determining the BC you want to be guided by the problem you are solving.
To illustrate, assume you are computing the probability that a stochastic process (for example, the position of a particle, the price of a stock, or other quantity represented by your stochastic process) will remain within a channel bounded by two values (or walls, or stock price levels) as time goes by. Here you may assume that the particle (or stock price) either vanishes or bounces when it touches one of the walls. If your particle vanishes when it touches a boundary, it means the probability of finding the particle "at" the boundary is zero - in this case, a Dirichlet BC of zero probability at that boundary is the right thing to do. Such BC will cause probability to drain from the solution domain as time goes by. If the particle bounces when it touches a wall, then your BC must reflect probability flow preservation.
Checking the probability flux numerically will give you an indication of how well your BC are doing (and also how well you are capturing your IC.) If your problem is such that whatever you are solving for doesn't disappear nor gets created in the solution domain, your probability flux must be numerically preserved.
Thanks Domingo Tavella
Very nice clarification of the subject. your answer really helps me.
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