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.^[1]

Definition

Given a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$ 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 $X$ be ${\mathcal {F}}_{T}$ 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 $X$ is given by ${\displaystyle \mathbb {E} ^{g}[X]:=Y_{0}}$. 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 ${\displaystyle m\times d}$ 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 $1$ function is the indicator function).^[1]

Existence and uniqueness

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 ${\mathcal {F}}_{t}$-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 $| \cdot |$ is a norm in $\mathbb {R} ^{m}$)
3. $g$ is Lipschitz continuous in $(y,z)$, 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 $C$

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 ${\mathcal {F}}_{t}$-adapted processes ${\displaystyle (Y,Z)}$ which satisfy the stochastic differential equation.^[2]

In particular, if $g$ additionally satisfies:

1. $g$ is continuous in time ($t$)
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 $t$.^[3]

See also

• 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]

References

1. Philippe Briand; François Coquet; Ying Hu; Jean Mémin; Shige Peng (2000). "A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation" (pdf). Electronic Communications in Probability. 5 (13): 101–117. doi:10.1214/ecp.v5-1025. Retrieved August 2, 2012.
2. Peng, S. (2004). "Nonlinear Expectations, Nonlinear Evaluations and Risk Measures". Stochastic Methods in Finance (pdf). Lecture Notes in Mathematics. 1856. pp. 165–138. doi:10.1007/978-3-540-44644-6_4. ISBN 978-3-540-22953-7. Retrieved August 9, 2012.
3. Chen, Z.; Chen, T.; Davison, M. (2005). "Choquet expectation and Peng's g -expectation". The Annals of Probability. 33 (3): 1179. doi:10.1214/009117904000001053.
4. Rosazza Gianin, E. (2006). "Risk measures via g-expectations". Insurance: Mathematics and Economics. 39: 19–65. doi:10.1016/j.insmatheco.2006.01.002.