G-expectation


In probability theory, the g-expectation is a nonlinear expectation based on a backwards stochastic differential equation originally developed by Shige Peng.

Definition

Given a probability space with is a Wiener process. 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 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 .

Existence and uniqueness

Let satisfy:
  1. is an -adapted process for every
  2. the L2 space
  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.