Law of supremum of time-scaled Brownian motion

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I would like to know if there is a formula for the law of
$$
sup_l leq t leq u fracB_tsqrtt
$$
where $B$ is a standard Brownian motion, and $0 < l < u < 1$ are constants?



The law of $sup_l leq t leq u B_t$ itself is well-known, but I couldn't find in any textbook at hand the law of the quantity that I am interested in..



Thanks for any help in advance!




stochastic-processes brownian-motion




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  • 1




    I think there is a problem at $0$. Brownian motion is not Holder continuous with exponent $1/2$.
    – Michael
    Aug 1 at 12:32










  • So the $sup$ should be infinite.
    – Michael
    Aug 1 at 12:32










  • @Michael, you are right. I carelessly put 0 and 1, which is not really the case in my work.
    – Dormire
    Aug 1 at 13:40














up vote
3
down vote

favorite
1












I would like to know if there is a formula for the law of
$$
sup_l leq t leq u fracB_tsqrtt
$$
where $B$ is a standard Brownian motion, and $0 < l < u < 1$ are constants?



The law of $sup_l leq t leq u B_t$ itself is well-known, but I couldn't find in any textbook at hand the law of the quantity that I am interested in..



Thanks for any help in advance!




stochastic-processes brownian-motion




share|cite|improve this question

















  • 1




    I think there is a problem at $0$. Brownian motion is not Holder continuous with exponent $1/2$.
    – Michael
    Aug 1 at 12:32










  • So the $sup$ should be infinite.
    – Michael
    Aug 1 at 12:32










  • @Michael, you are right. I carelessly put 0 and 1, which is not really the case in my work.
    – Dormire
    Aug 1 at 13:40












up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





I would like to know if there is a formula for the law of
$$
sup_l leq t leq u fracB_tsqrtt
$$
where $B$ is a standard Brownian motion, and $0 < l < u < 1$ are constants?



The law of $sup_l leq t leq u B_t$ itself is well-known, but I couldn't find in any textbook at hand the law of the quantity that I am interested in..



Thanks for any help in advance!




stochastic-processes brownian-motion




share|cite|improve this question













I would like to know if there is a formula for the law of
$$
sup_l leq t leq u fracB_tsqrtt
$$
where $B$ is a standard Brownian motion, and $0 < l < u < 1$ are constants?



The law of $sup_l leq t leq u B_t$ itself is well-known, but I couldn't find in any textbook at hand the law of the quantity that I am interested in..



Thanks for any help in advance!





stochastic-processes brownian-motion





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 1 at 14:27









Math1000

18.4k31444




18.4k31444









asked Aug 1 at 12:19









Dormire

479213




479213







  • 1




    I think there is a problem at $0$. Brownian motion is not Holder continuous with exponent $1/2$.
    – Michael
    Aug 1 at 12:32










  • So the $sup$ should be infinite.
    – Michael
    Aug 1 at 12:32










  • @Michael, you are right. I carelessly put 0 and 1, which is not really the case in my work.
    – Dormire
    Aug 1 at 13:40












  • 1




    I think there is a problem at $0$. Brownian motion is not Holder continuous with exponent $1/2$.
    – Michael
    Aug 1 at 12:32










  • So the $sup$ should be infinite.
    – Michael
    Aug 1 at 12:32










  • @Michael, you are right. I carelessly put 0 and 1, which is not really the case in my work.
    – Dormire
    Aug 1 at 13:40







1




1




I think there is a problem at $0$. Brownian motion is not Holder continuous with exponent $1/2$.
– Michael
Aug 1 at 12:32




I think there is a problem at $0$. Brownian motion is not Holder continuous with exponent $1/2$.
– Michael
Aug 1 at 12:32












So the $sup$ should be infinite.
– Michael
Aug 1 at 12:32




So the $sup$ should be infinite.
– Michael
Aug 1 at 12:32












@Michael, you are right. I carelessly put 0 and 1, which is not really the case in my work.
– Dormire
Aug 1 at 13:40




@Michael, you are right. I carelessly put 0 and 1, which is not really the case in my work.
– Dormire
Aug 1 at 13:40















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