In probability theory, the g-expectation is a nonlinear expectation based on a backwards stochastic differential equation (BSDE) originally developed by Shige Peng.
[1] Given a probability space
(
,
)
(
t
{\displaystyle (W_{t})_{t\geq 0}}
is a (d-dimensional) Wiener process (on that space).
Given the filtration generated by
= σ (
measurable.
Consider the BSDE given by: Then the g-expectation for
Note that if
is an m-dimensional vector, then
(for each time
) is an m-dimensional vector and
matrix.
In fact the conditional expectation is given by
and much like the formal definition for conditional expectation it follows that
function is the indicator function).
satisfy: Then for any random variable
there exists a unique pair of
-adapted processes
which satisfy the stochastic differential equation.
additionally satisfies: then for the terminal random variable
it follows that the solution processes
are square integrable.
is square integrable for all times