In this question, and this question, it is clear that the solution of the one-dimensional SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ is $X_t-X_0=B_t$. This is also the Bessel process. I am confused by the solution, as it is equivalent ti $d X_t = dB_t$. How can there be two equivalent SDE representations of the same stochastic process $X_t=W_t$?
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@saz: Can you please help me with this one? Thanks for your answers on SDE's! – Bravo Feb 22 '21 at 20:48
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I am not quite sure your conclusion that $X_t - X_0 = B_t$ is correct. It looks like you have $X_t^2 = (X_0 + B_t)^2$, but that doesn't imply $X_t = X_0 + B_t$. In fact, if $dX_t = \frac 1{2X_t}dt + dB_t$ then $X$ can't even be a local martingale. – user6247850 Feb 22 '21 at 23:43