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.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given the SDE: $dX_{t}=\sqrt{X_{t}}dW_{t},$ $\ X_{0}=1$ , where $W_{t}$ is a 1-d Brownian motion.

I was told that this SDE has a unique strong solution, but I don't know how to construct it. I know that this SDE has strong uniqueness, therefore I only need to construct a weak solution. I'm guessing we need to first consider a weak solution up to an explosion time (i.e. the hitting time of $X_{t}$ to the level $0$), but how to show such a weak solution exists?

Thank you for your help!

share|cite|improve this question
Check this: $X_t = (\frac{1}{2}(W_t-W_0)+1)^2$ – Nikita Evseev Oct 19 '12 at 4:21
@nikita2 This would satisfy $\mathrm{d}X_t = \frac{1}{4} \mathrm{d}t + \sqrt{X_t} \mathrm{d}W_t$. – Sasha Oct 19 '12 at 4:39
@Sasha Ok I just thought about PDE with separated variables. – Nikita Evseev Oct 19 '12 at 4:43
Wouldn't a solution to this SDE be negative for some $t$ with positive probability? And isn't that a problem when looking at $\sqrt{X_t}$? – Stefan Hansen Oct 22 '12 at 11:54
How do you know a strong solution exists? $\sqrt{x}$ is not Lipschitz at $0$. – Michael Jul 12 '14 at 11:19

Your Answer


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

Browse other questions tagged or ask your own question.