In probability theory , the g-expectation is a nonlinear expectation based on a backwards stochastic differential equation (BSDE) originally developed by Shige Peng .[1]
Definition [ ]
Given a probability space
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
with
(
W
t
)
t
≥
0
{\displaystyle (W_{t})_{t\geq 0}}
is a (d -dimensional) Wiener process (on that space). Given the filtration generated by
(
W
t
)
{\displaystyle (W_{t})}
, i.e.
F
t
=
σ
(
W
s
:
s
∈
[
0
,
t
]
)
{\displaystyle {\mathcal {F}}_{t}=\sigma (W_{s}:s\in [0,t])}
, let
X
{\displaystyle X}
be
F
T
{\displaystyle {\mathcal {F}}_{T}}
measurable . Consider the BSDE given by:
d
Y
t
=
g
(
t
,
Y
t
,
Z
t
)
d
t
−
Z
t
d
W
t
Y
T
=
X
{\displaystyle {\begin{aligned}dY_{t}&=g(t,Y_{t},Z_{t})\,dt-Z_{t}\,dW_{t}\\Y_{T}&=X\end{aligned}}}
Then the g-expectation for
X
{\displaystyle X}
is given by
E
g
[
X
]
:=
Y
0
{\displaystyle \mathbb {E} ^{g}[X]:=Y_{0}}
. Note that if
X
{\displaystyle X}
is an m -dimensional vector, then
Y
t
{\displaystyle Y_{t}}
(for each time
t
{\displaystyle t}
) is an m -dimensional vector and
Z
t
{\displaystyle Z_{t}}
is an
m
×
d
{\displaystyle m\times d}
matrix.
In fact the conditional expectation is given by
E
g
[
X
∣
F
t
]
:=
Y
t
{\displaystyle \mathbb {E} ^{g}[X\mid {\mathcal {F}}_{t}]:=Y_{t}}
and much like the formal definition for conditional expectation it follows that
E
g
[
1
A
E
g
[
X
∣
F
t
]
]
=
E
g
[
1
A
X
]
{\displaystyle \mathbb {E} ^{g}[1_{A}\mathbb {E} ^{g}[X\mid {\mathcal {F}}_{t}]]=\mathbb {E} ^{g}[1_{A}X]}
for any
A
∈
F
t
{\displaystyle A\in {\mathcal {F}}_{t}}
(and the
1
{\displaystyle 1}
function is the indicator function ).[1]
Existence and uniqueness [ ]
Let
g
:
[
0
,
T
]
×
R
m
×
R
m
×
d
→
R
m
{\displaystyle g:[0,T]\times \mathbb {R} ^{m}\times \mathbb {R} ^{m\times d}\to \mathbb {R} ^{m}}
satisfy:
g
(
⋅
,
y
,
z
)
{\displaystyle g(\cdot ,y,z)}
is an
F
t
{\displaystyle {\mathcal {F}}_{t}}
-adapted process for every
(
y
,
z
)
∈
R
m
×
R
m
×
d
{\displaystyle (y,z)\in \mathbb {R} ^{m}\times \mathbb {R} ^{m\times d}}
∫
0
T
|
g
(
t
,
0
,
0
)
|
d
t
∈
L
2
(
Ω
,
F
T
,
P
)
{\displaystyle \int _{0}^{T}|g(t,0,0)|\,dt\in L^{2}(\Omega ,{\mathcal {F}}_{T},\mathbb {P} )}
the L2 space (where
|
⋅
|
{\displaystyle |\cdot |}
is a norm in
R
m
{\displaystyle \mathbb {R} ^{m}}
)
g
{\displaystyle g}
is Lipschitz continuous in
(
y
,
z
)
{\displaystyle (y,z)}
, i.e. for every
y
1
,
y
2
∈
R
m
{\displaystyle y_{1},y_{2}\in \mathbb {R} ^{m}}
and
z
1
,
z
2
∈
R
m
×
d
{\displaystyle z_{1},z_{2}\in \mathbb {R} ^{m\times d}}
it follows that
|
g
(
t
,
y
1
,
z
1
)
−
g
(
t
,
y
2
,
z
2
)
|
≤
C
(
|
y
1
−
y
2
|
+
|
z
1
−
z
2
|
)
{\displaystyle |g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|\leq C(|y_{1}-y_{2}|+|z_{1}-z_{2}|)}
for some constant
C
{\displaystyle C}
Then for any random variable
X
∈
L
2
(
Ω
,
F
t
,
P
;
R
m
)
{\displaystyle X\in L^{2}(\Omega ,{\mathcal {F}}_{t},\mathbb {P} ;\mathbb {R} ^{m})}
there exists a unique pair of
F
t
{\displaystyle {\mathcal {F}}_{t}}
-adapted processes
(
Y
,
Z
)
{\displaystyle (Y,Z)}
which satisfy the stochastic differential equation.[2]
In particular, if
g
{\displaystyle g}
additionally satisfies:
g
{\displaystyle g}
is continuous in time (
t
{\displaystyle t}
)
g
(
t
,
y
,
0
)
≡
0
{\displaystyle g(t,y,0)\equiv 0}
for all
(
t
,
y
)
∈
[
0
,
T
]
×
R
m
{\displaystyle (t,y)\in [0,T]\times \mathbb {R} ^{m}}
then for the terminal random variable
X
∈
L
2
(
Ω
,
F
t
,
P
;
R
m
)
{\displaystyle X\in L^{2}(\Omega ,{\mathcal {F}}_{t},\mathbb {P} ;\mathbb {R} ^{m})}
it follows that the solution processes
(
Y
,
Z
)
{\displaystyle (Y,Z)}
are square integrable. Therefore
E
g
[
X
|
F
t
]
{\displaystyle \mathbb {E} ^{g}[X|{\mathcal {F}}_{t}]}
is square integrable for all times
t
{\displaystyle t}
.[3]
See also [ ]
Expected value
Choquet expectation
Risk measure – almost any time consistent convex risk measure can be written as
ρ
g
(
X
)
:=
E
g
[
−
X
]
{\displaystyle \rho _{g}(X):=\mathbb {E} ^{g}[-X]}
[4]
References [ ]
^ a b Philippe Briand; François Coquet; Ying Hu; Jean Mémin; Shige Peng (2000). "A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation" (PDF) . Electronic Communications in Probability . 5 (13): 101–117. doi :10.1214/ecp.v5-1025 . Archived from the original (pdf) on March 4, 2016. Retrieved August 2, 2012 .
^ Peng, S. (2004). "Nonlinear Expectations, Nonlinear Evaluations and Risk Measures". Stochastic Methods in Finance (PDF) . Lecture Notes in Mathematics. Vol. 1856. pp. 165–138. doi :10.1007/978-3-540-44644-6_4 . ISBN 978-3-540-22953-7 . Archived from the original (pdf) on March 3, 2016. Retrieved August 9, 2012 .
^ Chen, Z.; Chen, T.; Davison, M. (2005). "Choquet expectation and Peng's g -expectation". The Annals of Probability . 33 (3): 1179. arXiv :math/0506598 . doi :10.1214/009117904000001053 .
^ Rosazza Gianin, E. (2006). "Risk measures via g-expectations". Insurance: Mathematics and Economics . 39 : 19–65. doi :10.1016/j.insmatheco.2006.01.002 .