Stochastic Calculus - Limit Law

[danago]

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

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 :)

 

[danago]

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:

<br /> \int^{∞}_0 x^{\alpha} exp(-\frac{\beta}{x}) dx<br />

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?).

 

[danago]

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

<br /> \int^{∞}_0 x^{\alpha} exp(-\frac{\beta}{x}) dx = \beta^{\alpha+1}\Gamma(-\alpha-1)<br />

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