G-expectation

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

1. ${\displaystyle g(\cdot ,y,z)}$ is an ${\displaystyle {\mathcal {F}}_{t}}$-adapted process for every ${\displaystyle (y,z)\in \mathbb {R} ^{m}\times \mathbb {R} ^{m\times d}}$
2. ${\displaystyle \int _{0}^{T}|g(t,0,0)|\,dt\in L^{2}(\Omega ,{\mathcal {F}}_{T},\mathbb {P} )}$ the L2 space (where ${\displaystyle |\cdot |}$ is a norm in ${\displaystyle \mathbb {R} ^{m}}$)
3. ${\displaystyle g}$ is Lipschitz continuous in ${\displaystyle (y,z)}$, i.e. for every ${\displaystyle y_{1},y_{2}\in \mathbb {R} ^{m}}$ and ${\displaystyle z_{1},z_{2}\in \mathbb {R} ^{m\times d}}$ it follows that ${\displaystyle |g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|\leq C(|y_{1}-y_{2}|+|z_{1}-z_{2}|)}$ for some constant ${\displaystyle C}$

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

In particular, if ${\displaystyle g}$ additionally satisfies:

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

then for the terminal random variable ${\displaystyle X\in L^{2}(\Omega ,{\mathcal {F}}_{t},\mathbb {P}$ ;\mathbb {R} ^{m})} it follows that the solution processes ${\displaystyle (Y,Z)}$ are square integrable. Therefore ${\displaystyle \mathbb {E} ^{g}[X|{\mathcal {F}}_{t}]}$ is square integrable for all times ${\displaystyle t}$.^[3]

• Expected value
• Choquet expectation
• Risk measure – almost any time consistent convex risk measure can be written as ${\displaystyle \rho _{g}(X):=\mathbb {E} ^{g}[-X]}$^[4]