G-expectation

Given a probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$ with ${\displaystyle (W_{t})_{t\geq 0}}$ is a (d-dimensional) Wiener process (on that space). Given the filtration generated by ${\displaystyle (W_{t})}$, i.e. ${\displaystyle {\mathcal {F}}_{t}=\sigma (W_{s}:s\in [0,t])}$, let ${\displaystyle X}$ be ${\displaystyle {\mathcal {F}}_{T}}$ measurable. Consider the BSDE given by:

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

Then the g-expectation for ${\displaystyle X}$ is given by ${\displaystyle \mathbb {E} ^{g}[X]:=Y_{0}}$. Note that if ${\displaystyle X}$ is an m-dimensional vector, then ${\displaystyle Y_{t}}$ (for each time ${\displaystyle t}$) is an m-dimensional vector and ${\displaystyle Z_{t}}$ is an ${\displaystyle m\times d}$ matrix.

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

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]