ito lemma in finance

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 applying Ito’s lema the following formulas are useful:

$$ dW_tdW_t=dt $$
$$ dt_tdW_t=0 $$
$$ dtdt=0 $$

Applications

it’s used for example to derive Black-Scholes equation [in partial derivatives]