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 $(\mathrm{\Omega },ℱ,\mathbb{P})$ 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:

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

Then the g-expectation for $X$ is given by ${𝔼}^g[X]:=Y_0$. 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 ${𝔼}^g[X\mid {ℱ}_t]:=Y_t$ and much like the formal definition for conditional expectation it follows that ${𝔼}^g[1_A{𝔼}^g[X\mid {ℱ}_t]]={𝔼}^g[1_{A}X]$ for any $A\in {ℱ}_t$ (and the $1$ function is the indicator function).

Existence and uniqueness

Let $g:[0,T]\times {ℝ}^m\times {ℝ}^{m\times d}\to {ℝ}^m$ satisfy:

1. $g(\cdot ,y,z)$ is an ${ℱ}_t$-adapted process for every $(y,z)\in {ℝ}^m\times {ℝ}^{m\times d}$
2. $\underset{0}{\overset{T}{\int }}|g(t,0,0)|dt\in L^2(\mathrm{\Omega },{ℱ}_T,\mathbb{P})$ the L2 space (where $|\cdot |$ is a norm in ${ℝ}^m$)
3. $g$ is Lipschitz continuous in $(y,z)$, i.e. for every $y_1,y_2\in {ℝ}^m$ and $z_1,z_2\in {ℝ}^{m\times d}$ 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 $X\in L^2(\mathrm{\Omega },{ℱ}_t,\mathbb{P};{ℝ}^m)$ 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. $g(t,y,0)\equiv 0$ for all $(t,y)\in [0,T]\times {ℝ}^m$

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

See also

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