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Density of the Hitting Time of Brownian Motion — MeetProba
Exercises
/
Martingales & Stopping Times
Density of the Hitting Time of Brownian Motion
Density of the Hitting Time of Brownian Motion
Medium
Problems
Mark Solved
Exercise 3 / 7
Problem
Hints
(2)
Solution
Answer
a
2
π
t
exp
(
−
a
2
2
t
)
\frac{a}{\sqrt{2\pi t}} \exp\left(-\frac{a^2}{2t}\right)
2
π
t
a
exp
(
−
2
t
a
2
)
∣
a
∣
2
π
exp
(
−
a
2
2
t
)
\frac{|a|}{\sqrt{2\pi}} \exp\left(-\frac{a^2}{2t}\right)
2
π
∣
a
∣
exp
(
−
2
t
a
2
)
a
2
2
π
t
3
exp
(
−
a
2
2
t
)
\frac{a^2}{\sqrt{2\pi t^3}} \exp\left(-\frac{a^2}{2t}\right)
2
π
t
3
a
2
exp
(
−
2
t
a
2
)
a
2
π
t
3
exp
(
−
a
2
2
t
)
\frac{a}{\sqrt{2\pi t^3}} \exp\left(-\frac{a^2}{2t}\right)
2
π
t
3
a
exp
(
−
2
t
a
2
)
Submit
Let
(
W
t
)
t
≥
0
(W_t)_{t \geq 0}
(
W
t
)
t
≥
0
be a standard Brownian motion and
τ
a
:
=
inf
{
t
≥
0
∣
W
t
=
a
}
,
a
>
0.
\tau_a := \inf\{t \geq 0 \mid W_t = a\}, \qquad a > 0.
τ
a
:=
in
f
{
t
≥
0
∣
W
t
=
a
}
,
a
>
0.
Questions:
Compute the cumulative distribution function (CDF):
F
τ
a
(
t
)
=
P
(
τ
a
≤
t
)
.
F_{\tau_a}(t) = \mathbb{P}(\tau_a \leq t).
F
τ
a
(
t
)
=
P
(
τ
a
≤
Deduce the probability density function (PDF) of
τ
a
\tau_a
τ
a
.
t
)
.