Let $X_t=X_0+M_t+A_t$ a continuous semi-martingale. Let $g: \Bbb R \to [-1,1]$, of class $C^{\infty}$, with $g(x)= \left\{ \begin{matrix} -1, & x \le 0 \\ 1, & x \ge 1 \end{matrix} \right.$. Let $f_n : \Bbb R \to \Bbb R$ with $f_n(0)=0$ and $f_n'(x)=g(nx) \ \forall x \in \Bbb R$.
Show $\lim \limits _{n \to \infty} \sup \limits _{t \ge 0} | \int \limits _0 ^t [\text{sgn} (X_s) - f_n' (X_s)] \Bbb d M_s| = 0$ in $L^2(\Omega,P)$ and $\lim \limits _{n \to \infty} \sup \limits _{t \ge 0} |\int \limits _0 ^t [\text{sgn} (X_s) - f_n' (X_s)] \Bbb d A_s| = 0$, where $\text{sgn} (x) = \left\{ \begin{matrix} 1, & x \ge 0 \\ -1, & x<0 \end{matrix} \right.$.
How to do these two questions? Thank you.