G-expectation explained

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, l{F}, P ) with (W _{t)} _{t} is a (d-dimensional) Wiener process (on that space). Given the filtration generated by (W _{t)} , i.e. l{F} _{t} = \sigma(W _{s:} s \in [ 0 ,t]) , let X be l{F} _{T} measurable. Consider the BSDE given by:\begin{align}dY _{t} & = g(t,Y _{t,Z} _{t)} dt - Z _{t} dW _{t\ Y} _{T} & = X\end{align} 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 x d matrix.

In fact the conditional expectation is given by

E ^{g[X} \mid l{F} _{t]} : = Y _{t} and much like the formal definition for conditional expectation it follows that E ^{g[ 1} _{A} E ^{g[X} \mid l{F} _{t]]} = E ^{g[ 1} _{A} X] for any A \in l{F} _{t} (and the 1 function is the indicator function).

Existence and uniqueness

Let

g: [ 0 ,T] x R ^{m} x R ^{m} \to R ^{m} satisfy:g( ⋅ ,y,z) is an l{F} _{t} -adapted process for every (y,z) \in R ^{m} x R ^{m} \int ^{T} _{0} |g(t, 0 , 0 )| dt \in L ^{2 (\Omega, l{F}} _{T, P )} the L2 space (where | ⋅ | is a norm in R ^{m} )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} 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 ^{m)} _{t, P ; R} there exists a unique pair of l{F} _{t} -adapted processes (Y,Z) which satisfy the stochastic differential equation.^[2]

In particular, if

g additionally satisfies:g is continuous in time (t )g(t,y, 0 ) \equiv 0 for all (t,y) \in [ 0 ,T] x R ^{m} then for the terminal random variable X \in L ^{m)} _{t, P ; R} it follows that the solution processes (Y,Z) are square integrable. Therefore E ^{g[X} | l{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

\rho _{g(X)} : = E ^{g[-X]} ^[4]

Notes and References

1. A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation . Philippe Briand . François Coquet . Ying Hu . Jean Mémin . Shige Peng . Electronic Communications in Probability . 2000 . 5 . 13 . 101–117 .
2. Book: Peng . S. . Nonlinear Expectations, Nonlinear Evaluations and Risk Measures . 10.1007/978-3-540-44644-6_4 . Stochastic Methods in Finance . Lecture Notes in Mathematics . 1856 . 165–138 . 2004 . 978-3-540-22953-7 . pdf . August 9, 2012 . https://web.archive.org/web/20160303223024/http://sisla06.samsi.info/fmse/pm/ShigePeng.pdf . March 3, 2016 . dead .
3. Chen . Z. . Chen . T. . Davison . M. . 10.1214/009117904000001053 . Choquet expectation and Peng's g -expectation . The Annals of Probability . 33 . 3 . 1179 . 2005 . math/0506598 .
4. Rosazza Gianin . E. . 10.1016/j.insmatheco.2006.01.002 . Risk measures via g-expectations . Insurance: Mathematics and Economics . 39 . 19–65 . 2006 .