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.

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:

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).

Existence and uniqueness

Let satisfy:

1. is an -adapted process for every
2. the L2 space (where is a norm in )
3. is Lipschitz continuous in , i.e. for every and it follows that for some constant

Then for any random variable there exists a unique pair of -adapted processes which satisfy the stochastic differential equation.

In particular, if additionally satisfies:

1. is continuous in time ()
2. for all

then for the terminal random variable it follows that the solution processes are square integrable. Therefore is square integrable for all times .

See also

• Expected value
• Choquet expectation
• Risk measure – almost any time consistent convex risk measure can be written as

References