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Stochastic Calculus - Limit Law

  1. Oct 31, 2011 #1

    danago

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    [PLAIN]http://img17.imageshack.us/img17/1061/stochcalcq4.png [Broken]

    I am currently taking a class in quantitative finance, part of which includes an introduction to stochastic calculus. This is the first time i have encountered stochastic differential equations, so it is all quite new to me. I am going ok with most of it, however i got stuck on the question shown above. I did part a without any issues by showing that dx<0 whenever x is greater than the long term value, that dx>0 when it is less, and dx=0 when it is equal to the long term value.

    I am stuck, however, on part b; i am just not sure how to approach it. I thought about actually trying to solve it for X(t), but made no progress with that. A push in the right direction would be much appreciated :)
     
    Last edited by a moderator: May 5, 2017
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  3. Nov 1, 2011 #2

    danago

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    I have gone through my lecture notes again and I may have made some progress. Solving the forward time-homogenous Kolmogorov-Chapman equation gives an expression for the density function, whose integral on (0,∞) should be convergent. I have ended up with an integral of the form:

    [tex]
    \int^{∞}_0 x^{\alpha} exp(-\frac{\beta}{x}) dx
    [/tex]

    Where α and β are just combinations of the parameters in the question which i have grouped for simplicity. It has turned out that α<-2 does indeed lead to the required condition, σ2<2a. I am not sure however why α<-2 does imply the convergence of the integral (or is it just a coincidence that it gives me the criterion that i want?).
     
  4. Nov 1, 2011 #3

    danago

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    If i evaluate the integral in the previous post, i get:

    [tex]
    \int^{∞}_0 x^{\alpha} exp(-\frac{\beta}{x}) dx = \beta^{\alpha+1}\Gamma(-\alpha-1)
    [/tex]

    I am still not sure how to logically get the inequality that i want though.
     
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