G-expectation

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