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
with
is a (d-dimensional)
Wiener process (on that space). Given the
filtration generated by
, i.e.
, let
be
measurable. Consider the BSDE given by:
![{\displaystyle {\begin{aligned}dY_{t}&=g(t,Y_{t},Z_{t})\,dt-Z_{t}\,dW_{t}\\Y_{T}&=X\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72361697fa5931eebb2fc44e4c583da6802485bf)
Then the g-expectation for
is given by
. Note that if
is an m-dimensional vector, then
(for each time
) is an m-dimensional vector and
is an
matrix.
In fact the
conditional expectation is given by
and much like the formal definition for conditional expectation it follows that
for any
(and the
function is the
indicator function).
[1]
Existence and uniqueness
Let
satisfy:
is an
-
adapted process for every ![{\displaystyle (y,z)\in \mathbb {R} ^{m}\times \mathbb {R} ^{m\times d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a482f239892bff06c0d82e6603954dac01fa7af)
the
L2 space (where
is a norm in
)
is
Lipschitz continuous in
, i.e. for every
and
it follows that
for some constant ![{\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
Then for any random variable
there exists a unique pair of
-adapted processes
which satisfy the stochastic differential equation.
[2]
In particular, if
additionally satisfies:
is continuous in time (
)
for all ![{\displaystyle (t,y)\in [0,T]\times \mathbb {R} ^{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a81acf0f98d9e357333b4986538397979576a2)
then for the terminal random variable
it follows that the solution processes
are square integrable. Therefore
is square integrable for all times
.
[3]
See also
References