Different Forms of Itô’s Lemma

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

Case 1a: F = F(X)

dF=12d2FdX2dt+dFdXdXdF = \frac{1}{2}\frac{d^2F}{dX^2}dt + \frac{dF}{dX}dX

Case 1b: F = F(t, X)

dF=(Ft+122FX2)dt+FXdXdF = \left(\frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial X^2} \right)dt + \frac{\partial F}{\partial X}dX

Case 2: Function is in terms of Z

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

Case 2a: F = F(Z)

dF=(adFdZ+12b2d2FdZ2)dt+(bdFdZ)dXdF = \left(a\frac{dF}{dZ} + \frac{1}{2}b^2\frac{d^2F}{dZ^2} \right)dt + \left(b\frac{dF}{dZ} \right)dX

Case 2b: F = F(t, Z)

dF=(Ft+aFZ+12b22FZ2)dt+(bFZ)dXdF = \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

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