to derive equations for asset price evolution one uses Ito lemma: $$ d f(X_t,Y_t) = f_{x}(X)dX+f_{y}(Y)dY+f_{xy}dXdY + \frac{1}{2} f_{xx}(X) dXdX+ \frac{1}{2} f_{yy}(Y) dY dY $$ basically its the same as Taylor formula for 2 variables developed until 2nd order. when…

to change the probability measure one uses Girsanov theorem (formula): $$ \frac{N_a(0)}{N_a(T)} d \mathbb{P}_a=\frac{N_b(0)}{N_b(T)} d \mathbb{P}_b$$ $$N_a$$ is numeraire (price of any non dividend paying asset, usually bond or bank account) and it’s correspoding probability measure is $$P_a$$ $$N_b$$ is…