### PENALISATIONS OF BROWNIAN MOTION WITH ITS MAXIMUM AND MINIMUM PROCESSES AS WEAK FORMS OF SKOROKHOD EMBEDDING

**B. ROYNETTE, P. VALLOIS, AND M. YOR**

*Theory of Stochastic Processes*

*Vol. 14 (30), no. 2, 2008, pp. 116–138*

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 $F_t$ is the indicator function of $\{I_t\geq \alpha, \ S_t\leq \beta\}$ or $F_t$ 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.

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