Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider the following SDE

$$ X_t = x + \int_0^t \theta (\mu -X_s) ds + \int_0^t\kappa \sqrt{|X_s|} dW_s $$

where W is a brownian motion. I'm trying to show a weak solution exists, does anyone have a tip to get me started?

share|improve this question
    
Perhaps, a good book on SDEs such as Revuz and Yor? Also, this may be of a help. –  Ilya May 22 '13 at 11:17
    
Thanks for your interest, i did indeed solve it myself. But I will check out your references anyway :) –  Henrik May 22 '13 at 12:38

1 Answer 1

up vote 1 down vote accepted

The result is most about knowing which theorem to use, so you guys didn't really stand a chance. Here is a proof:

The theorem proving weak existence is by skorokhod and states: Let $b$ denote the drift of the SDE and $\sigma$ the diffusion then if $$|\sigma(x)| + |b(x)| \leq c_1 + c_2|x|$$

that is they are of most linear growth, then there is weak existence (but not necessarily weak uniqueness).

We use that the solution to CIR is non-negative (this can be shown by a comparison result). Then notice that $b(x)=\theta\cdot \mu - \theta x$ and $\sigma (x)\leq \kappa + \kappa \cdot x$ since $\sqrt{x}\leq 1$ for $x \leq 1$ and after that $\sqrt{x} \leq x$. So choosing e.g. $c_1=\theta\cdot \mu + \kappa $ and $c_2 = \kappa - \theta \cdot \mu$ we are done.

As bonus info it is quite easy to prove strong uniqueness, if one then uses the fact that weak existence and strong uniqueness implies strong existence we got a unique strong solution to the CIR process.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.