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










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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















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










share|cite|improve this question

















  • 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













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










share|cite|improve this question













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






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asked Sep 10 at 20:31









Marco

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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













  • 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
















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