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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