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