G expectation

Definition

Given a probability space ( Ω , F , P ) with ( W t ) t ≥ 0 is a (d-dimensional) Wiener process (on that space). Given the filtration generated by ( W t ) , i.e. F t = σ ( W s : s ∈ [ 0 , t ] ) , let X be F T measurable. Consider the BSDE given by:

Then the g-expectation for X is given by E g [ X ] := Y 0 . Note that if X is an m-dimensional vector, then Y t (for each time t ) is an m-dimensional vector and Z t is an m × d matrix.

In fact the conditional expectation is given by E g [ X ∣ F t ] := Y t and much like the formal definition for conditional expectation it follows that E g [ 1 A E g [ X ∣ F t ] ] = E g [ 1 A X ] for any A ∈ F t (and the 1 function is the indicator function).

Existence and uniqueness

Let g : [ 0 , T ] × R m × R m × d → R m satisfy:

1. g ( ⋅ , y , z ) is an F t -adapted process for every ( y , z ) ∈ R m × R m × d
2. ∫ 0 T | g ( t , 0 , 0 ) | d t ∈ L 2 ( Ω , F T , P ) the L2 space (where | ⋅ | is a norm in R m )
3. g is Lipschitz continuous in ( y , z ) , i.e. for every y 1 , y 2 ∈ R m and z 1 , z 2 ∈ R m × d it follows that | g ( t , y 1 , z 1 ) − g ( t , y 2 , z 2 ) | ≤ C ( | y 1 − y 2 | + | z 1 − z 2 | ) for some constant C

Then for any random variable X ∈ L 2 ( Ω , F t , P ; R m ) there exists a unique pair of F t -adapted processes ( Y , Z ) which satisfy the stochastic differential equation.

In particular, if g additionally satisfies:

1. g is continuous in time ( t )
2. g ( t , y , 0 ) ≡ 0 for all ( t , y ) ∈ [ 0 , T ] × R m

then for the terminal random variable X ∈ L 2 ( Ω , F t , P ; R m ) it follows that the solution processes ( Y , Z ) are square integrable. Therefore E g [ X | F t ] is square integrable for all times t .