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I want to find $P[S_T≤K , M_T≥U]$, where $S_T$ is a geometric Brownian motion, $M_T$ its maximum over $[0,T]$ and $K,U$ two constants, as a function of $P[S_{T_1}≤K , M_{T_1}≥U]$ and $P[S_{T_2}≤K , M_{T_2}≥U]$ with $T_1<T_2<T$.

Any help?

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