Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 202 / 2 / e

By the Itô formula,To compute the Stratonovich integral correction, apply the Itô formula also to $f^{\prime }$, which is twice continuously differentiable because $f$ is $C^3$. The continuous local martingale part of $f^{\prime }(X)$ is $\int f^{\prime \prime }(X_s)d{L}_s$, where $L$ is the continuous local martingale part of $X$. A finite-variation process has zero quadratic covariation with $X$, and the quadratic variation of a stochastic integral together with polarization identity givesAll these statements can be localized to compact ranges of $X$, so unbounded derivatives create no global integrability requirement. Substitute this identity into the Stratonovich integral to obtain the Stratonovich chain rule:This is the integrated meaning of the requested differential identity; there is no extra second-order term after the Stratonovich integral correction has been included.