Throughout let $(,$ F$,$ P$)$ be a given probability space

Assume that a filtration $(F_t{)}_t=0^T$ is given. Find an example of a financial market

model S$($S with hat$)$ and a trading strategy on all assets $(H^0,H)$ such that

$1.(H^0,H)$ is self$-$financing and is not identically null;

$2.(H^0,H)$ is not self$-$financing.