Let \(X\) and \(Y\) be continuous semi-martingales. The Stratonovich integral of \(X\) w.r.t. \(Y\) may be defined as

\[\int_{0}^{t} X_{s} \circ d Y_{s}=\int_{0}^{t} X_{s} d Y_{s}+\frac{1}{2}\langle X, Yangle_{t}\]

Prove that

\[\int_{0}^{t} X_{s} \circ d Y_{s}=(u c p) \lim _{n \rightarrow \infty} \sum_{i=0}^{p(n)-1}\left(\frac{X_{t_{i}^{n}}+X_{t_{i+1}^{n}}}{2}\right)\left(Y_{t_{i+1}^{n}}-Y_{t_{i}^{n}}\right)\]

where \(0=t_{0}

\[f\left(X_{t}\right)=f\left(X_{0}\right)+\int_{0}^{t} f^{\prime}\left(X_{s}\right) \circ d X_{s}\]

For a Brownian motion, the Stratonovich integral may also be approximated as

\[\int_{0}^{t} \varphi\left(B_{s}\right) \circ d B_{s}=\lim _{n \rightarrow \infty} \sum_{i=0}^{p(n)-1} \varphi\left(B_{\left(t_{i}+t_{i+1}\right) / 2}\right)\left(B_{t_{i+1}}-B_{t_{i}}\right)\]

where the limit is in probability; however, such an approximation does not hold in general for continuous semi-martingales. Assumption on \(f\) in the integral form of \(f\left(X_{t}\right)\). The Stratonovich integral can be extended to general semimartingales (not necessarily continuous).