Different Forms of Itô’s Lemma

Chen — Sun, 10 Oct 2021 11:38:26 +0000

Case 1: Function is in terms of X (Brownian Motion)

dF = \frac{1}{2}\frac{d^2F}{dX^2}dt + \frac{dF}{dX}dX

dF = \left(\frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial X^2} \right)dt + \frac{\partial F}{\partial X}dX

dZ = a(Z, t) dt + b(Z, t) dX

dF = \left(a\frac{dF}{dZ} + \frac{1}{2}b^2\frac{d^2F}{dZ^2} \right)dt + \left(b\frac{dF}{dZ} \right)dX

dF = \left(\frac{\partial F}{\partial t} + a\frac{\partial F}{\partial Z} + \frac{1}{2}b^2\frac{\partial^2 F}{\partial Z^2} \right)dt + \left(b\frac{\partial F}{\partial Z} \right)dX