0.1 Itô Formula
For an Itô Process that satisfies the stochastic differential equation \[ dX_t=\mu_tdt+\sigma_tdB_t \] where \(B_t\) is a [[wiener-process-brownian-motion]], for a twice-differentiable scalar function \(f(t,x)\) of two real variables \(t\) and \(x\) one has \[ df(t,X_t)=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}dX_t+\frac{1}{2}\frac{\partial^2f}{\partial x^2}dX_t^2 \] where \(dX_t^2\) denotes the quadratic variation of \(X_t\).
We evaluate \((dX_{t})^2 = (dX_{t})\cdot (dX_{t})\) is computed according to the rules \[ dt\cdot dt = dt\cdot dB_{t}=dB_{t}\cdot dt=0,\quad dB_{t}\cdot dB_{t}=dt. \]