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Expected truncated hitting time of Brownian motion
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I'm trying to compute the expected value of a truncated hitting time of a Brownian motion with an one-sided boundary. More specifically, if we let $B_t$ be a $(mu, sigma)$ Brownian motion and $T= inf{tge 0: B_t = a}$ be a hitting time for some $a>0, I want to know if there is any closed form formula for the expected truncated hitting time
$$
mathbb{E}[tau] = mathbb{E}[T_a wedge t_0]
$$
where $t_0>0$ is a constant. I found out that $T_a$ follows the inverse Gaussian distribution, but I was not able to obtain a closed form solution for the truncated hitting time.
probability stochastic-processes martingales
asked Jan 26 at 19:30
user_causer_ca
13
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add a comment |
$begingroup$
I'm trying to compute the expected value of a truncated hitting time of a Brownian motion with an one-sided boundary. More specifically, if we let $B_t$ be a $(mu, sigma)$ Brownian motion and $T= inf{tge 0: B_t = a}$ be a hitting time for some $a>0, I want to know if there is any closed form formula for the expected truncated hitting time
$$
mathbb{E}[tau] = mathbb{E}[T_a wedge t_0]
$$
where $t_0>0$ is a constant. I found out that $T_a$ follows the inverse Gaussian distribution, but I was not able to obtain a closed form solution for the truncated hitting time.
probability stochastic-processes martingales
asked Jan 26 at 19:30
user_causer_ca
13
$endgroup$
add a comment |
$begingroup$
I'm trying to compute the expected value of a truncated hitting time of a Brownian motion with an one-sided boundary. More specifically, if we let $B_t$ be a $(mu, sigma)$ Brownian motion and $T= inf{tge 0: B_t = a}$ be a hitting time for some $a>0, I want to know if there is any closed form formula for the expected truncated hitting time
$$
mathbb{E}[tau] = mathbb{E}[T_a wedge t_0]
$$
where $t_0>0$ is a constant. I found out that $T_a$ follows the inverse Gaussian distribution, but I was not able to obtain a closed form solution for the truncated hitting time.
probability stochastic-processes martingales
asked Jan 26 at 19:30
user_causer_ca
13
$endgroup$
I'm trying to compute the expected value of a truncated hitting time of a Brownian motion with an one-sided boundary. More specifically, if we let $B_t$ be a $(mu, sigma)$ Brownian motion and $T= inf{tge 0: B_t = a}$ be a hitting time for some $a>0, I want to know if there is any closed form formula for the expected truncated hitting time
$$
mathbb{E}[tau] = mathbb{E}[T_a wedge t_0]
$$
where $t_0>0$ is a constant. I found out that $T_a$ follows the inverse Gaussian distribution, but I was not able to obtain a closed form solution for the truncated hitting time.
probability stochastic-processes martingales
probability stochastic-processes martingales
asked Jan 26 at 19:30
user_causer_ca
13
asked Jan 26 at 19:30
user_causer_ca
13
asked Jan 26 at 19:30
user_causer_ca
13
asked Jan 26 at 19:30
user_causer_ca
13
asked Jan 26 at 19:30
user_causer_ca
13
13
add a comment |
add a comment |
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