Polynomial Itô formula from integration by parts

The one-dimensional Itô formula for polynomials follows by induction on powers using the stochastic product rule. If it holds for $X^k$, its martingale part gives $[X^k,X]=k{X}^{k-1}\cdot [X]$. The product rule for $X^{k}X$ then yields the coefficient $k(k+1)/2$ of the bracket term for $X^{k+1}$.

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