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 $$(\Omega,\mathcal{F},\mathbb{P})$$ with $$(W_t)_{t \geq 0}$$ is a (d-dimensional) Wiener process (on that space). Given the filtration generated by $$(W_t)$$, i.e. $$\mathcal{F}_t = \sigma(W_s: s \in [0,t])$$, let $$X$$ be $$\mathcal{F}_T$$ measurable. Consider the BSDE given by:
 * $$ \begin{align}dY_t &= g(t,Y_t,Z_t) \, dt - Z_t \, dW_t\\ Y_T &= X\end{align}$$

Then the g-expectation for $$X$$ is given by $$\mathbb{E}^g[X] := Y_0$$. 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 $$m \times d$$ matrix.

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

Existence and uniqueness
Let $$g: [0,T] \times \mathbb{R}^m \times \mathbb{R}^{m \times d} \to \mathbb{R}^m$$ satisfy: Then for any random variable $$X \in L^2(\Omega,\mathcal{F}_t,\mathbb{P};\mathbb{R}^m)$$ there exists a unique pair of $$\mathcal{F}_t$$-adapted processes $$(Y,Z)$$ which satisfy the stochastic differential equation.
 * 1) $$g(\cdot,y,z)$$ is an $$\mathcal{F}_t$$-adapted process for every $$(y,z) \in \mathbb{R}^m \times \mathbb{R}^{m \times d}$$
 * 2) $$\int_0^T |g(t,0,0)| \, dt \in L^2(\Omega,\mathcal{F}_T,\mathbb{P})$$ the L2 space (where $$| \cdot |$$ is a norm in $$\mathbb{R}^m$$)
 * 3) $$g$$ is Lipschitz continuous in $$(y,z)$$, i.e. for every $$y_1,y_2 \in \mathbb{R}^m$$ and $$z_1,z_2 \in \mathbb{R}^{m \times d}$$ it follows that $$|g(t,y_1,z_1) - g(t,y_2,z_2)| \leq C (|y_1 - y_2| + |z_1 - z_2|)$$ for some constant $$C$$

In particular, if $$g$$ additionally satisfies: then for the terminal random variable $$X \in L^2(\Omega,\mathcal{F}_t,\mathbb{P};\mathbb{R}^m)$$ it follows that the solution processes $$(Y,Z)$$ are square integrable. Therefore $$\mathbb{E}^g[X | \mathcal{F}_t]$$ is square integrable for all times $$t$$.
 * 1) $$g$$ is continuous in time ($$t$$)
 * 2) $$g(t,y,0) \equiv 0$$ for all $$(t,y) \in [0,T] \times \mathbb{R}^m$$