Regularity requirement for convergence of Euler scheme for stochastic integral?

Question

Let $S_t$ be follow Black Scholes, then I am interesting in simulating the process

$\int ^t _0 e^{-rt}1_{\{S_t\leq K\}}dS_t$

which is like a naive hedge of a European put, which does not work in practice.

1. Am I correct to say no Milstein Type scheme exists due to discontinuous derivative
2. Does the Euler scheme produce a process which approximate this stochastic integral?