Brownian bridge and first hitting times

• Tetef
However, in summary, the conversation discusses the definition of first hitting times in a standard brownian motion and the probability of one hitting time being before another. The questioner is seeking help in finding this probability in the case of a brownian bridge. They suggest a possible solution involving probability density functions.
Tetef
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 :
$P\{T_{a}<T_{b}\}=\frac{b}{b-a}$

I'm looking for this probability in the case of a brownian bridge :
$P\{T_{a}<T_{b} | W(t)=x\}$ with $x<a$

Could some one help me please?

Thx !

Your question looks confusing. Your last statement has x < a. Also did you mean W(0) = x? Your T definitions use t also.

I did mean what I wrote,

A standard brownian motion has $W(0)=0$

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 $P\{T_{a}<T_{b}|W(t)=x\}$, $t$ is the 'time of one observation'.
If you look for $P\{T_{a}<t\}$, you're looking for the probability for $W$ to hit a level $a$ before $t$ ($t$ being the end of the observation).

Here, in the expression $P\{T_{a}<T_{b}|W(t)=x\}$, I'm looking for the probability for one hitting time to be before the other, knowing that at the time $t$, the brownian motion will be equal to $x$, i.e. $W(t)=x$; and thus form a brownian bridge between $0$ at time $0$ ($W(0)=0$) and $x$ at time $t$ ($W(t)=x$)

$P\{T_{a}<T_{b}|W(t)=x\}$ could be asked for $x\inℝ$.
In my case I have $x<a$. This implies, by continuity of the brownian motion, that $T_{a}$ does exist before $t$. It might make it simpler, it might not...

Last edited:
I don't know if it's any use but, I've thought of this :

$P\{T_{a}<T_{b}|W(t)=x\}=\frac{p\{T_{a}<T_{b}\cap W(t)=x\}}{p\{W(t)=x\}}$

where capital $P$ are probabilities and small $p$ are probability density functions.

we know :
$p\{W(t)=x\}=\frac{1}{\sqrt{2\pi t}}exp( -\frac{x^2}{2t})$

we don't know :
$p\{T_{a}<T_{b}\cap W(t)=x\}$

but we know that :
$\int^{+∞}_{-∞} p\{T_{a}<T_{b}\cap W(t)=x\}dx=\frac{b}{b-a}$
Is that right?

I have not studied this problem in any detail, so I don't know if I can be of any further help.

What is Brownian bridge?

Brownian bridge is a mathematical concept used to describe the movement of a particle or molecule in a fluid environment. It is based on the Brownian motion theory, which states that small particles in a fluid will exhibit random, erratic movements due to collisions with other particles. Brownian bridge specifically refers to the path of a particle between two specific points in time, with the assumption that the particle starts and ends at rest.

How is Brownian bridge related to first hitting times?

First hitting times is a concept that is often studied in conjunction with Brownian bridge. It refers to the time it takes for a particle to reach a specific position or boundary for the first time. In the context of Brownian bridge, first hitting times can be used to analyze the likelihood of a particle reaching a certain point or crossing a certain boundary within a given time frame.

What are some real-world applications of Brownian bridge and first hitting times?

Brownian bridge and first hitting times have many practical applications in various fields such as physics, chemistry, finance, and biology. In physics, they can be used to study the behavior of particles in a fluid environment. In finance, they can be used to model stock prices and predict market movements. In biology, they can be used to analyze the movement of molecules in a cell.

Can Brownian bridge and first hitting times be used to predict future events?

While Brownian bridge and first hitting times can provide valuable insights into the behavior of particles or molecules, they cannot be used to accurately predict future events. This is because they are based on random, probabilistic movements and cannot account for all variables that may affect the outcome. They can, however, provide useful statistical data and trends that can inform decision-making processes.

Are there any limitations to the use of Brownian bridge and first hitting times?

Like any mathematical model, Brownian bridge and first hitting times have their limitations. They are based on simplifying assumptions and may not accurately reflect real-world scenarios. Additionally, they may not be applicable to all types of particles or fluids. It is important to carefully consider the assumptions and limitations when applying these concepts to a specific situation.

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