- Source: Reflection principle (Wiener process)
In the theory of probability for stochastic processes, the reflection principle for a Wiener process states that if the path of a Wiener process f(t) reaches a value f(s) = a at time t = s, then the subsequent path after time s has the same distribution as the reflection of the subsequent path about the value a. More formally, the reflection principle refers to a lemma concerning the distribution of the supremum of the Wiener process, or Brownian motion. The result relates the distribution of the supremum of Brownian motion up to time t to the distribution of the process at time t. It is a corollary of the strong Markov property of Brownian motion.
Statement
If
(
W
(
t
)
:
t
≥
0
)
{\displaystyle (W(t):t\geq 0)}
is a Wiener process, and
a
>
0
{\displaystyle a>0}
is a threshold (also called a crossing point), then the lemma states:
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
)
=
2
P
(
W
(
t
)
≥
a
)
{\displaystyle \mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)=2\mathbb {P} (W(t)\geq a)}
Assuming
W
(
0
)
=
0
{\displaystyle W(0)=0}
, due to the continuity of Wiener processes, each path (one sampled realization) of Wiener process on
(
0
,
t
)
{\displaystyle (0,t)}
which finishes at or above value/level/threshold/crossing point
a
{\displaystyle a}
the time
t
{\displaystyle t}
(
W
(
t
)
≥
a
{\displaystyle W(t)\geq a}
) must have crossed (reached) a threshold
a
{\displaystyle a}
(
W
(
t
a
)
=
a
{\displaystyle W(t_{a})=a}
) at some earlier time
t
a
≤
t
{\displaystyle t_{a}\leq t}
for the first time . (It can cross level
a
{\displaystyle a}
multiple times on the interval
(
0
,
t
)
{\displaystyle (0,t)}
, we take the earliest.)
For every such path, you can define another path
W
′
(
t
)
{\displaystyle W'(t)}
on
(
0
,
t
)
{\displaystyle (0,t)}
that is reflected or vertically flipped on the sub-interval
(
t
a
,
t
)
{\displaystyle (t_{a},t)}
symmetrically around level
a
{\displaystyle a}
from the original path. These reflected paths are also samples of the Wiener process reaching value
W
′
(
t
a
)
=
a
{\displaystyle W'(t_{a})=a}
on the interval
(
0
,
t
)
{\displaystyle (0,t)}
, but finish below
a
{\displaystyle a}
. Thus, of all the paths that reach
a
{\displaystyle a}
on the interval
(
0
,
t
)
{\displaystyle (0,t)}
, half will finish below
a
{\displaystyle a}
, and half will finish above. Hence, the probability of finishing above
a
{\displaystyle a}
is half that of reaching
a
{\displaystyle a}
.
In a stronger form, the reflection principle says that if
τ
{\displaystyle \tau }
is a stopping time then the reflection of the Wiener process starting at
τ
{\displaystyle \tau }
, denoted
(
W
τ
(
t
)
:
t
≥
0
)
{\displaystyle (W^{\tau }(t):t\geq 0)}
, is also a Wiener process, where:
W
τ
(
t
)
=
W
(
t
)
χ
{
t
≤
τ
}
+
(
2
W
(
τ
)
−
W
(
t
)
)
χ
{
t
>
τ
}
{\displaystyle W^{\tau }(t)=W(t)\chi _{\left\{t\leq \tau \right\}}+(2W(\tau )-W(t))\chi _{\left\{t>\tau \right\}}}
and the indicator function
χ
{
t
≤
τ
}
=
{
1
,
if
t
≤
τ
0
,
otherwise
{\displaystyle \chi _{\{t\leq \tau \}}={\begin{cases}1,&{\text{if }}t\leq \tau \\0,&{\text{otherwise }}\end{cases}}}
and
χ
{
t
>
τ
}
{\displaystyle \chi _{\{t>\tau \}}}
is defined similarly. The stronger form implies the original lemma by choosing
τ
=
inf
{
t
≥
0
:
W
(
t
)
=
a
}
{\displaystyle \tau =\inf \left\{t\geq 0:W(t)=a\right\}}
.
Proof
The earliest stopping time for reaching crossing point a,
τ
a
:=
inf
{
t
:
W
(
t
)
=
a
}
{\displaystyle \tau _{a}:=\inf \left\{t:W(t)=a\right\}}
, is an almost surely bounded stopping time. Then we can apply the strong Markov property to deduce that a relative path subsequent to
τ
a
{\displaystyle \tau _{a}}
, given by
X
t
:=
W
(
t
+
τ
a
)
−
a
{\displaystyle X_{t}:=W(t+\tau _{a})-a}
, is also simple Brownian motion independent of
F
τ
a
W
{\displaystyle {\mathcal {F}}_{\tau _{a}}^{W}}
. Then the probability distribution for the last time
W
(
s
)
{\displaystyle W(s)}
is at or above the threshold
a
{\displaystyle a}
in the time interval
[
0
,
t
]
{\displaystyle [0,t]}
can be decomposed as
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
)
=
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
,
W
(
t
)
≥
a
)
+
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
,
W
(
t
)
<
a
)
=
P
(
W
(
t
)
≥
a
)
+
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
,
X
(
t
−
τ
a
)
<
0
)
{\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)\geq a\right)+\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)
.
By the tower property for conditional expectations, the second term reduces to:
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
,
X
(
t
−
τ
a
)
<
0
)
=
E
[
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
,
X
(
t
−
τ
a
)
<
0
|
F
τ
a
W
)
]
=
E
[
χ
sup
0
≤
s
≤
t
W
(
s
)
≥
a
P
(
X
(
t
−
τ
a
)
<
0
|
F
τ
a
W
)
]
=
1
2
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
)
,
{\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,X(t-\tau _{a})<0\right)&=\mathbb {E} \left[\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,X(t-\tau _{a})<0|{\mathcal {F}}_{\tau _{a}}^{W}\right)\right]\\&=\mathbb {E} \left[\chi _{\sup _{0\leq s\leq t}W(s)\geq a}\mathbb {P} \left(X(t-\tau _{a})<0|{\mathcal {F}}_{\tau _{a}}^{W}\right)\right]\\&={\frac {1}{2}}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right),\end{aligned}}}
since
X
(
t
)
{\displaystyle X(t)}
is a standard Brownian motion independent of
F
τ
a
W
{\displaystyle {\mathcal {F}}_{\tau _{a}}^{W}}
and has probability
1
/
2
{\displaystyle 1/2}
of being less than
0
{\displaystyle 0}
. The proof of the lemma is completed by substituting this into the second line of the first equation.
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
)
=
P
(
W
(
t
)
≥
a
)
+
1
2
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
)
P
(
sup
0
≤
s
≤
t
W
(
s
)
≥
a
)
=
2
P
(
W
(
t
)
≥
a
)
{\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=\mathbb {P} \left(W(t)\geq a\right)+{\frac {1}{2}}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)\\\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=2\mathbb {P} \left(W(t)\geq a\right)\end{aligned}}}
.
Consequences
The reflection principle is often used to simplify distributional properties of Brownian motion. Considering Brownian motion on the restricted interval
(
W
(
t
)
:
t
∈
[
0
,
1
]
)
{\displaystyle (W(t):t\in [0,1])}
then the reflection principle allows us to prove that the location of the maxima
t
max
{\displaystyle t_{\text{max}}}
, satisfying
W
(
t
max
)
=
sup
0
≤
s
≤
1
W
(
s
)
{\displaystyle W(t_{\text{max}})=\sup _{0\leq s\leq 1}W(s)}
, has the arcsine distribution. This is one of the Lévy arcsine laws.
References
Kata Kunci Pencarian:
- Reflection principle (Wiener process)
- Reflection principle (disambiguation)
- Norbert Wiener
- Diffusion process
- Autoregressive model
- Gaussian random field
- Dartmouth workshop
- SABR volatility model
- Fourier transform
- Ouroboros
