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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?

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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
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.

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