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 (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{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 \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).[1]

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.[2]

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.[3]

See also

References

  1. ↑ 1.0 1.1 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.