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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
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 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Snell_envelope&oldid=888355863 "
C a t e g o r y :
● M a t h e m a t i c a l f i n a n c e
H i d d e n c a t e g o r i e s :
● A r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s f r o m A p r i l 2 0 1 2
● A l l a r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s
● T h i s p a g e w a s l a s t e d i t e d o n 1 8 M a r c h 2 0 1 9 , a t 1 6 : 2 6 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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