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The Snell envelope , used in stochastics and mathematical finance , is the smallest supermartingale dominating a stochastic process . The Snell envelope is named after James Laurie Snell .
Definition [ edit ]
Given a filtered probability space
(
Ω
,
F
,
(
F
t
)
t
∈
[
0
,
T
]
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in [0,T]},\mathbb {P} )}
and an absolutely continuous probability measure
Q
≪
P
{\displaystyle \mathbb {Q} \ll \mathbb {P} }
then an adapted process
U
=
(
U
t
)
t
∈
[
0
,
T
]
{\displaystyle U=(U_{t})_{t\in [0,T]}}
is the Snell envelope with respect to
Q
{\displaystyle \mathbb {Q} }
of the process
X
=
(
X
t
)
t
∈
[
0
,
T
]
{\displaystyle X=(X_{t})_{t\in [0,T]}}
if
U
{\displaystyle U}
is a
Q
{\displaystyle \mathbb {Q} }
-supermartingale
U
{\displaystyle U}
dominates
X
{\displaystyle X}
, i.e.
U
t
≥
X
t
{\displaystyle U_{t}\geq X_{t}}
Q
{\displaystyle \mathbb {Q} }
-almost surely for all times
t
∈
[
0
,
T
]
{\displaystyle t\in [0,T]}
If
V
=
(
V
t
)
t
∈
[
0
,
T
]
{\displaystyle V=(V_{t})_{t\in [0,T]}}
is a
Q
{\displaystyle \mathbb {Q} }
-supermartingale which dominates
X
{\displaystyle X}
, then
V
{\displaystyle V}
dominates
U
{\displaystyle U}
.[1]
Construction [ edit ]
Given a (discrete) filtered probability space
(
Ω
,
F
,
(
F
n
)
n
=
0
N
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n=0}^{N},\mathbb {P} )}
and an absolutely continuous probability measure
Q
≪
P
{\displaystyle \mathbb {Q} \ll \mathbb {P} }
then the Snell envelope
(
U
n
)
n
=
0
N
{\displaystyle (U_{n})_{n=0}^{N}}
with respect to
Q
{\displaystyle \mathbb {Q} }
of the process
(
X
n
)
n
=
0
N
{\displaystyle (X_{n})_{n=0}^{N}}
is given by the recursive scheme
U
N
:=
X
N
,
{\displaystyle U_{N}:=X_{N},}
U
n
:=
X
n
∨
E
Q
[
U
n
+
1
∣
F
n
]
{\displaystyle U_{n}:=X_{n}\lor \mathbb {E} ^{\mathbb {Q} }[U_{n+1}\mid {\mathcal {F}}_{n}]}
for
n
=
N
−
1
,
.
.
.
,
0
{\displaystyle n=N-1,...,0}
where
∨
{\displaystyle \lor }
is the join (in this case equal to the maximum of the two random variables).[1]
Application [ edit ]
If
X
{\displaystyle X}
is a discounted American option payoff with Snell envelope
U
{\displaystyle U}
then
U
t
{\displaystyle U_{t}}
is the minimal capital requirement to hedge
X
{\displaystyle X}
from time
t
{\displaystyle t}
to the expiration date.[1]
References [ edit ]
^ a b c Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 280–282. ISBN 9783110183467 .