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

$\begin{array}{rl}d{Y}_t& =g(t,Y_t,Z_t)dt-Z_{t}d{W}_t\\ Y_T& =X\end{array}$

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 ${\mathbb{𝔼}}^g[X\mid {{ℱ}}_t]:=Y_t$ and much like the formal definition for conditional expectation it follows that ${\mathbb{𝔼}}^g[1_A{\mathbb{𝔼}}^g[X\mid {{ℱ}}_t]]={\mathbb{𝔼}}^g[1_{A}X]$ for any $A\in {{ℱ}}_t$ (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:

1. $g(\cdot ,y,z)$ is an ${{ℱ}}_t$-adapted process for every $(y,z)\in {\mathbb{ℝ}}^m\times {\mathbb{ℝ}}^{m\times d}$
2. $\underset{0}{\overset{T}{\int }}|g(t,0,0)|dt\in L^2(\mathrm{\Omega },{{ℱ}}_T,\mathbb{\mathbb{P}})$ 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 $|g(t,y_1,z_1)-g(t,y_2,z_2)|\le C(|y_1-y_2|+|z_1-z_2|)$ for some constant $C$

Then for any random variable $X\in L^2(\mathrm{\Omega },{{ℱ}}_t,\mathbb{\mathbb{P}};{\mathbb{ℝ}}^m)$ 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. $g(t,y,0)\equiv 0$ for all $(t,y)\in [0,T]\times {\mathbb{ℝ}}^m$

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

See also

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