Proving weak existence of CIR process

Question

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?

Answer 1

I was also looking for an answer, I was going to ask Henrik for the source of the cited theorem by Skorokhod (because I couldn't find it), but being new to this forum, I couldn't leave a comment. So, here's what I think.

Many sources refer to Revuz and Yor, Section IX.3, but there's no existence result there. I think that's because the remaining step is trivial (not to me but to trained minds). Consider processes $\{x^n\}$ defined by $${\rm d} x_t^n =[(\theta(\mu -x_t^n))\vee(-n)]\,{\rm d}t+ ((\kappa\sqrt{x_t^n})\wedge n)\,{\rm d}w_t.$$ (For simplicity, I'm assuming $\mu>0$. $\theta$ should be positive, and there's no log in assuming $\kappa>0$.) Skorokhod's theorem (the well-known one assuming bounded continuous coefficients) implies weak existence, and Yamada and Watanabe's theorem implies pathwise uniqueness, which then, combined, imply the existence of a unique strong solution (as Henrik mentioned). What remains, I think, is to define the stopping times $$\tau_n=\inf\{t\ge 0:\theta(\mu-x_t^n)=-n\text{ or }\kappa\sqrt{x_t^n}=n\}$$ and note $x_t^n=x_t^m$ for all $t\le\tau_n$ and all $m\ge n$; then define $x_t=x_t^n$ on $\{t\le\tau_n\}$. This $x$ satisfies the original SDE (up to explosion if there is one, or indefinitely otherwise).