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 [math]\displaystyle{ (\Omega,\mathcal{F},\mathbb{P}) }[/math] with [math]\displaystyle{ (W_t)_{t \geq 0} }[/math] is a (d-dimensional) Wiener process (on that space). Given the filtration generated by [math]\displaystyle{ (W_t) }[/math], i.e. [math]\displaystyle{ \mathcal{F}_t = \sigma(W_s: s \in [0,t]) }[/math], let [math]\displaystyle{ X }[/math] be [math]\displaystyle{ \mathcal{F}_T }[/math] measurable. Consider the BSDE given by:

[math]\displaystyle{ \begin{align}dY_t &= g(t,Y_t,Z_t) \, dt - Z_t \, dW_t\\ Y_T &= X\end{align} }[/math]

Then the g-expectation for [math]\displaystyle{ X }[/math] is given by [math]\displaystyle{ \mathbb{E}^g[X] := Y_0 }[/math]. Note that if [math]\displaystyle{ X }[/math] is an m-dimensional vector, then [math]\displaystyle{ Y_t }[/math] (for each time [math]\displaystyle{ t }[/math]) is an m-dimensional vector and [math]\displaystyle{ Z_t }[/math] is an [math]\displaystyle{ m \times d }[/math] matrix.

In fact the conditional expectation is given by [math]\displaystyle{ \mathbb{E}^g[X \mid \mathcal{F}_t] := Y_t }[/math] and much like the formal definition for conditional expectation it follows that [math]\displaystyle{ \mathbb{E}^g[1_A \mathbb{E}^g[X \mid \mathcal{F}_t]] = \mathbb{E}^g[1_A X] }[/math] for any [math]\displaystyle{ A \in \mathcal{F}_t }[/math] (and the [math]\displaystyle{ 1 }[/math] function is the indicator function).

Existence and uniqueness

Let [math]\displaystyle{ g: [0,T] \times \mathbb{R}^m \times \mathbb{R}^{m \times d} \to \mathbb{R}^m }[/math] satisfy:

1. [math]\displaystyle{ g(\cdot,y,z) }[/math] is an [math]\displaystyle{ \mathcal{F}_t }[/math]-adapted process for every [math]\displaystyle{ (y,z) \in \mathbb{R}^m \times \mathbb{R}^{m \times d} }[/math]
2. [math]\displaystyle{ \int_0^T |g(t,0,0)| \, dt \in L^2(\Omega,\mathcal{F}_T,\mathbb{P}) }[/math] the L2 space (where [math]\displaystyle{ | \cdot | }[/math] is a norm in [math]\displaystyle{ \mathbb{R}^m }[/math])
3. [math]\displaystyle{ g }[/math] is Lipschitz continuous in [math]\displaystyle{ (y,z) }[/math], i.e. for every [math]\displaystyle{ y_1,y_2 \in \mathbb{R}^m }[/math] and [math]\displaystyle{ z_1,z_2 \in \mathbb{R}^{m \times d} }[/math] it follows that [math]\displaystyle{ |g(t,y_1,z_1) - g(t,y_2,z_2)| \leq C (|y_1 - y_2| + |z_1 - z_2|) }[/math] for some constant [math]\displaystyle{ C }[/math]

Then for any random variable [math]\displaystyle{ X \in L^2(\Omega,\mathcal{F}_t,\mathbb{P};\mathbb{R}^m) }[/math] there exists a unique pair of [math]\displaystyle{ \mathcal{F}_t }[/math]-adapted processes [math]\displaystyle{ (Y,Z) }[/math] which satisfy the stochastic differential equation.

In particular, if [math]\displaystyle{ g }[/math] additionally satisfies:

1. [math]\displaystyle{ g }[/math] is continuous in time ([math]\displaystyle{ t }[/math])
2. [math]\displaystyle{ g(t,y,0) \equiv 0 }[/math] for all [math]\displaystyle{ (t,y) \in [0,T] \times \mathbb{R}^m }[/math]

then for the terminal random variable [math]\displaystyle{ X \in L^2(\Omega,\mathcal{F}_t,\mathbb{P};\mathbb{R}^m) }[/math] it follows that the solution processes [math]\displaystyle{ (Y,Z) }[/math] are square integrable. Therefore [math]\displaystyle{ \mathbb{E}^g[X | \mathcal{F}_t] }[/math] is square integrable for all times [math]\displaystyle{ t }[/math].

See also

• Expected value
• Choquet expectation
• Risk measure – almost any time consistent convex risk measure can be written as [math]\displaystyle{ \rho_g(X) := \mathbb{E}^g[-X] }[/math]