I am reading a paper in which the author applies Ito's lemma to show that a particular process is a local martingale.
Given an SDE $dZ(t)=\frac{a_i^+(Z(t))}{Z(t)}dt+\sqrt{a_i^+(Z(t))}dW(t)$, the author computes the Stochastic differential of $M(t):=\log Z(t)$ using Ito's lemma. and claims that the stochastic differential of $M(t)$ is given by
$dM(t)=\frac{\sqrt{a_i^+(Z(t))}}{Z(t)}dW(t)$ He then goes ahead and uses a theorem on Local martingales to make conclusions on the process $M(t)$ which is crucial in the proof.
Shouldn't the stochastic differential be $dM(t)=\frac{\sqrt{a_i^+(Z(t))}}{Z(t)}dW(t)+\frac{1}{2} \frac{a_i^+(Z(t))}{Z(t)}{Z(t)^2}dt$ instead? I am also attaching the screenshot of the paper
You can find the paper here https://link.springer.com/article/10.1007/s12044-013-0141-8( the text just under equation (3.5) ) .
(Note that there is $a$ instead of $a_i$ which is a typo among the many typos and mistakes in the paper.)
You are correct, the trend term only gets reduced by half. Reversing the calculation, starting with $Z=\exp(M)$, gives in the Ito formula $$ dZ=Z(dM+\frac12d\langle M\rangle ) $$ where the quadratic variation is $d⟨M⟩=\frac{a^+(Z)}{Z^2}dt$, so that the original equation should have been $$ dZ=\sqrt{a^+(Z)}dW + \frac12\frac{a^+(Z)}{Z}dt. $$ The question now is, is this a genuine error or was the factor 2 in the denominator lost in print?