# Brownian bridge and first hitting times

 P: 3 Hi, Letting $W$ be a standard brownian motion, we define the first hitting times $T_{a}=inf\{t:W(t)=a\}$ with $a<0$ and $T_{b}=inf\{t:W(t)=b\}$ with $b>0$ The probability of one hitting time being before an other is : $P\{T_{a}  Sci Advisor P: 6,070 Your question looks confusing. Your last statement has x < a. Also did you mean W(0) = x? Your T definitions use t also.  P: 3 I did mean what I wrote, A standard brownian motion has [itex]W(0)=0$ I use $t$ in the definition $T_{a}=inf\{t:W(t)=a\}$to say the first hitting times are defined as : $T_{a}$ is the smallest time $t$ where the brownian motion does hit the level $a$ i.e. $W(t)=a$ In the expression of the probability [itex]P\{T_{a}