Penalisations of brownian motion with its maximum and minimum processes as weak forms of Skorokhod embedding

We develop a Brownian penalisation procedure related to weight processes $(F_t)$ of the type: $F_t:=f(I_t, S_t)$ where $f$ is a bounded function with compact support and $S_t$ (resp. $I_t$) is the one-sided maximum (resp. minimum) of the Brownian motion up to time $t.$ Two main cases are treated: either $F_t$t is the indicator function of $\{I_t\geq\alpha, S_t\leq\beta\}$ or $F_t$t is null when $\{S_t-I_t>c\}$ for some $c>0.$ Then we apply these results to some kind of asymptotic Skorokhod embedding problem.