I was reading this paper and page 14 the model is given.

I'm trying to find the steps to get to the SDE given.

OI is the open interest, E is the elasticity of demand. $$ \frac{\Delta S}{S} ∼ E·Q^p \\ \frac{\Delta S}{S}∝ E· |\frac{D}{<V>}|^psign(D) \ \ \ \ \ \frac{D}{<V>}>> 1 $$ $$ D = - OI \frac{\partial \delta(S,t)}{\partial t}dt $$ $$where\ \delta = N(d1)$$ I have also found the simplified notation $$ \frac{dS_t}{S_t} = EQdt + \sigma dW_t\\ Qdt = OI \frac{\partial \delta}{\partial \tau} dt,\ \ \tau = T-t $$ and the sde found is $$ dy = -\frac{E·OI}{V} \frac{y-a(T-t)}{\sqrt{2\pi \sigma^2(T-t)^3}} e^{-\frac{(y+a(T-t))^2}{2\sigma^2(T-t)}} dt+ \sigma dW $$ where $$y = ln(S/K),\ a = \mu + \frac{\sigma^2}{2}$$

From what I understand one could start by applying Ito to f = y = ln(S/K), with $$ \frac{\partial f}{\partial S} = 1/S, \ \frac{\partial^2 f}{\partial S^2} = -1/S^2$$ but in that case I'm not sure what the partial derivation would look like for $$ \frac{\partial f}{\partial t}dt$$


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