Paradox Brownian Motion : P(first passage time < infinite) = 1 yet E(first passage time) = infinite?
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I am studying the bible of stochastic calculus for finance by Shreve aka God.
But in the section "first passage time to level m" for the Brownian Motion there is a paradox :
1) P(first passage time < infinite) = 1
2) E(first passage time) = infinite
For me it is a paradox. How can you be at the same time finite and infinite?
Is there a solution or a interpretation to solve the paradox ?
Thanks
stochastic-calculus brownian-motion finance
asked Sep 10 at 20:31
Marco
41
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up vote
0
down vote
favorite
I am studying the bible of stochastic calculus for finance by Shreve aka God.
But in the section "first passage time to level m" for the Brownian Motion there is a paradox :
1) P(first passage time < infinite) = 1
2) E(first passage time) = infinite
For me it is a paradox. How can you be at the same time finite and infinite?
Is there a solution or a interpretation to solve the paradox ?
Thanks
stochastic-calculus brownian-motion finance
asked Sep 10 at 20:31
Marco
41
4
This is not a paradox: tons of random variables are 1. almost surely finite, and 2. not integrable. For a similar situation with no alea, consider $f(x)=1/x$ on $x>1$, then $f(x)$ is finite for every $x>1$ "but" $int_1^infty f(x)dx$ is infinite.
– Did
Sep 10 at 20:40
3
Take pdf $1/x^2$ in $[1,infty)$. Find the expected value.
– karakfa
Sep 10 at 20:43
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am studying the bible of stochastic calculus for finance by Shreve aka God.
But in the section "first passage time to level m" for the Brownian Motion there is a paradox :
1) P(first passage time < infinite) = 1
2) E(first passage time) = infinite
For me it is a paradox. How can you be at the same time finite and infinite?
Is there a solution or a interpretation to solve the paradox ?
Thanks
stochastic-calculus brownian-motion finance
asked Sep 10 at 20:31
Marco
41
I am studying the bible of stochastic calculus for finance by Shreve aka God.
But in the section "first passage time to level m" for the Brownian Motion there is a paradox :
1) P(first passage time < infinite) = 1
2) E(first passage time) = infinite
For me it is a paradox. How can you be at the same time finite and infinite?
Is there a solution or a interpretation to solve the paradox ?
Thanks
stochastic-calculus brownian-motion finance
stochastic-calculus brownian-motion finance
asked Sep 10 at 20:31
Marco
41
asked Sep 10 at 20:31
Marco
41
asked Sep 10 at 20:31
Marco
41
asked Sep 10 at 20:31
Marco
41
asked Sep 10 at 20:31
Marco
41
41
4
This is not a paradox: tons of random variables are 1. almost surely finite, and 2. not integrable. For a similar situation with no alea, consider $f(x)=1/x$ on $x>1$, then $f(x)$ is finite for every $x>1$ "but" $int_1^infty f(x)dx$ is infinite.
– Did
Sep 10 at 20:40
3
Take pdf $1/x^2$ in $[1,infty)$. Find the expected value.
– karakfa
Sep 10 at 20:43
add a comment |Â
4
This is not a paradox: tons of random variables are 1. almost surely finite, and 2. not integrable. For a similar situation with no alea, consider $f(x)=1/x$ on $x>1$, then $f(x)$ is finite for every $x>1$ "but" $int_1^infty f(x)dx$ is infinite.
– Did
Sep 10 at 20:40
3
Take pdf $1/x^2$ in $[1,infty)$. Find the expected value.
– karakfa
Sep 10 at 20:43
4
4
This is not a paradox: tons of random variables are 1. almost surely finite, and 2. not integrable. For a similar situation with no alea, consider $f(x)=1/x$ on $x>1$, then $f(x)$ is finite for every $x>1$ "but" $int_1^infty f(x)dx$ is infinite.
– Did
Sep 10 at 20:40
This is not a paradox: tons of random variables are 1. almost surely finite, and 2. not integrable. For a similar situation with no alea, consider $f(x)=1/x$ on $x>1$, then $f(x)$ is finite for every $x>1$ "but" $int_1^infty f(x)dx$ is infinite.
– Did
Sep 10 at 20:40
3
3
Take pdf $1/x^2$ in $[1,infty)$. Find the expected value.
– karakfa
Sep 10 at 20:43
Take pdf $1/x^2$ in $[1,infty)$. Find the expected value.
– karakfa
Sep 10 at 20:43
add a comment |Â
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4
This is not a paradox: tons of random variables are 1. almost surely finite, and 2. not integrable. For a similar situation with no alea, consider $f(x)=1/x$ on $x>1$, then $f(x)$ is finite for every $x>1$ "but" $int_1^infty f(x)dx$ is infinite.
– Did
Sep 10 at 20:40
3
Take pdf $1/x^2$ in $[1,infty)$. Find the expected value.
– karakfa
Sep 10 at 20:43