G-expectation

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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{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 {\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. Retrieved August 2, 2012. 
  2. ↑ Peng, S. (2004). "Nonlinear Expectations, Nonlinear Evaluations and Risk Measures" (pdf). Stochastic Methods in Finance. 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. 


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