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Letting [itex]W[/itex] be a standard brownian motion, we define the first hitting times

[itex]T_{a}=inf\{t:W(t)=a\}[/itex] with [itex]a<0[/itex]

and

[itex]T_{b}=inf\{t:W(t)=b\}[/itex] with [itex]b>0[/itex]

The probability of one hitting time being before an other is :

[itex]P\{T_{a}<T_{b}\}=\frac{b}{b-a}[/itex]

I'm looking for this probability in the case of a brownian bridge :

[itex]P\{T_{a}<T_{b} | W(t)=x\}[/itex] with [itex]x<a[/itex]

Could some one help me please?

Thx !

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# Brownian bridge and first hitting times

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