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

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Given a probability space   with   is a (d-dimensional) Wiener process (on that space). Given the filtration generated by  , i.e.  , let   be   measurable. Consider the BSDE given by:

 

Then the g-expectation for   is given by  . Note that if   is an m-dimensional vector, then   (for each time  ) is an m-dimensional vector and   is an   matrix.

In fact the conditional expectation is given by   and much like the formal definition for conditional expectation it follows that   for any   (and the   function is the indicator function).[1]

Existence and uniqueness

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Let   satisfy:

  1.   is an  -adapted process for every  
  2.   the L2 space (where   is a norm in  )
  3.   is Lipschitz continuous in  , i.e. for every   and   it follows that   for some constant  

Then for any random variable   there exists a unique pair of  -adapted processes   which satisfy the stochastic differential equation.[2]

In particular, if   additionally satisfies:

  1.   is continuous in time ( )
  2.   for all  

then for the terminal random variable   it follows that the solution processes   are square integrable. Therefore   is square integrable for all times  .[3]

See also

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References

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  1. ^ a b 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.
  2. ^ Peng, S. (2004). "Nonlinear Expectations, Nonlinear Evaluations and Risk Measures". Stochastic Methods in Finance (PDF). Lecture Notes in Mathematics. Vol. 1856. pp. 165–138. doi:10.1007/978-3-540-44644-6_4. ISBN 978-3-540-22953-7. Archived from the original (pdf) on March 3, 2016. 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. arXiv:math/0506598. 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.