If \( X \) and \( Y \) are continuous semimartingales, define the midpoint-integral as follows. For \( t \geq 0 \) fixed, let \( 0 = t_0^n < \cdots < t_{p_n}^n = t \) be a sequence of subdivisions with mesh \( \to 0 \). Define the following limit in probability: \[ \int_0^t X_s \circ dY_s = \lim_{n \to \infty} \sum_{i=0}^{p_n - 1} \left( \frac{X_{t_{i+1}^n} + X_{t_i^n}}{2} ight) \left( Y_{t_{i+1}^n} - Y_{t_i^n} ight). \] \begin{enumerate} \item[(a)] Show that this limit exists. \item[(b)] Prove that a.s., for all \( t \geq 0 \), \[ \int_0^t X_s \circ dY_s = \int_0^t X_s dY_s + \frac{1}{2} \langle X, Y angle_t, \] thereby establishing that the midpoint-integral has a modification with continuous paths. \item[(c)] If \( f : \mathbb{R} \to \mathbb{R} \) is \( C^3 \), show that a.s., for all \( t \geq 0 \), \[ f(X_t) = f(X_0) + \int_0^t f'(X_s) \circ dX_s. \]