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I am interested in the distribution of $Z(t) = \max\{B(t),m\}$ where $B(t)$ is a standard Brownian motion and $m$ is a constant. By distribution, I mean the distribution of $Z(t)$ for a given $t$. I tried some keywords to do a search but I am not even sure what it would be called.

Any direction on it would be appreciated. I am not sure where I should start and if there is any work done on the distribution of $Z(t)$.

marvinthemartian
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  • Which "distribution" do you mean? At points? Covariances between different times? – Ian Apr 28 '16 at 13:00
  • @Ian I meant the distribution at a given t. I also edited the question. – marvinthemartian Apr 28 '16 at 13:05
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    Notice that ${Z(t)\le x}={B(t)\le x}$ if $x\ge m$, while ${Z(t)\le x}=\emptyset$ if $x<m$. – John Dawkins Apr 28 '16 at 13:12
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    At a given $t$ the question is trivial, $Z(t)$ has the density that you get by cutting off the values below $m$ for a Gaussian with mean zero and variance $t$ and then renormalizing. So you have $\frac{e^{-\frac{x^2}{2t}}}{\int_m^\infty e^{-\frac{y^2}{2t}} dy}$ for $x \geq m$ and zero otherwise (I think I dealt with the constants correctly). – Ian Apr 28 '16 at 13:13
  • Would $Z(t)$ still be Brownian motion? Or is there a name for this type of process that I can look into? – marvinthemartian Apr 28 '16 at 13:25
  • $Z(t)$ isn't a Brownian motion, in fact it's not Markov. There might be a loose relation to the running maximum process $M(t)=\max_{s \in [0,t]} B(t)$ but it's not quite the same. – Ian Apr 29 '16 at 02:01

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