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Functions solving Chapman-Kolmogorov

  1. Mar 29, 2012 #1
    How many functions solve the Chapman-Kolmogorov equation? The Chapman-Kolmogorov equation is defined as:

    [tex]f({x_n}|{x_m}) = \int_{ - \infty }^{ + \infty } {f({x_n}|{x_o})} f({x_o}|{x_m})d{x_o}[/tex]

    And I've seen it proved that the following function will satisfy the above condition:

    [tex]f({x_n}|{x_o}) = {\left( {\frac{\lambda }{{2\pi ({t_n} - {t_0})}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
    \kern-0.1em/\kern-0.15em
    \lower0.25ex\hbox{$\scriptstyle 2$}}}}{e^{\frac{{\lambda {{({x_n} - {x_0})}^2}}}{{2({t_n} - {t_0})}}}}[/tex]

    But I have to wonder if there are any other functions that satisfy it. Any help is appreciated.
     
    Last edited: Mar 29, 2012
  2. jcsd
  3. Mar 30, 2012 #2
    I did find this solution to the Chapman-Kolmogorov equation:

    http://www.pims.math.ca/files/monahan_2b_0.pdf

    Although, I could not find any reference to the author. And I don 't know how general or particular his solution is. Maybe a more experienced person would like to take a look at it and advice. Thanks.
     
  4. Mar 30, 2012 #3

    chiro

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    I don't really know all that much about DE's (ODE's or PDE's) but that equation for joint probabilities looks pretty accurate.
     
  5. Jun 20, 2012 #4
    I believe just about any transition density for a markovian process should satisfy the C-K equation.
    The solution listed above was for a "raw" brownian motion. You can see that any arithmetic or geometric process will result in a similar process. The post above had a link to show that an Ornstein-Uhlenbeck (Langevin) process can satisfy it.
     
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