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:

${\begin{aligned}dY_{t}&=g(t,Y_{t},Z_{t})\,dt-Z_{t}\,dW_{t}\\Y_{T}&=X\end{aligned}}$

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:

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 $C$

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.

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 $X\in L^{2}(\Omega ,{\mathcal {F}}_{t},\mathbb {P} ;\mathbb {R} ^{m})$ 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 $t$.

See also

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