As a non-math student, I have been struggling studying stochastic calculus. Below is the first question I encounter and I have no map in my mind on how to approach this problem.

Let $ X_t $, $ Y_t $ be two continuous semimartingales. Given $ t\leq 0 $, let $ \mathcal{P}_n $ be a sequence of finite partitions over $ [0,t] $ such that mesh($ \mathcal{P}_n $)$ \rightarrow $ 0. Show that the limit: \begin{split} \int_0^t X_s\circ dY_s= \lim_{n\rightarrow\infty} \sum_{t_i\in \mathcal{P}_n} \frac{X_{t_{i-1}}+X_{t_i}}{2}(Y_{t_i}-Y_{t_{i-1}}) \end{split}

exists in probability, and the process $ t\mapsto \int_0^t X_s\circ dY_s$ is a modification of a continuous semimartingale. In addition, for $ f\in C^3(\mathbf{R}^1) $, show that:

\begin{split} f(X_t) = f(X_0)+\int_{0}^{t}f'(X_s)\circ dX_s \end{split}


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