A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation

Philippe Briand (Université Rennes 1)
François Coquet (Université Rennes 1)
Ying Hu (Université Rennes 1)
Jean Mémin (Université Rennes 1)
Shige Peng (Shandong University)


In [1], Z. Chen proved that, if for each terminal condition $\xi$, the solution of the BSDE associated to the standard parameter $(\xi, g_1)$ is equal at time $t=0$ to the solution of the BSDE associated to $(\xi, g_2)$ then we must have $g_1\equiv g_2$. This result yields a natural question: what happens in the case of an inequality in place of an equality? In this paper, we try to investigate this question and we prove some properties of ``$g$-expectation'', notion introduced by S. Peng in [8].

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Pages: 101-117

Publication Date: May 23, 2000

DOI: 10.1214/ECP.v5-1025


  1. Z. Chen, A property of backward stochastic differential equations, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 4, 483--488. Math. Review 99i:60116.
  2. N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1--71; Math. Review 98d:90030.
  3. H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, in Ecole d'e´ete´ de probabilités de Saint-Flour, XII---1982, 143--303, Lecture Notes in Math., 1097, Springer, Berlin, 1984. Math. Review 87m:60127.
  4. E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equations, Systems Control Lett. 14 (1990), no. 1, 55--61. Math. Review 91e:60171.
  5. E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic partial differential equations and their applications (Charlotte, NC, 1991), 200--217, Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992. Math. Review 93k:60157.
  6. S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep. 37 (1991), no. 1-2, 61--74. Math. Review 93a:35159.
  7. S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim. 30 (1992), no. 2, 284--304; Math. Review 93e:60123.
  8. S. Peng, Backward SDE and related g-expectation, in Backward stochastic differential equations, 141--159, Pitman res. Notes Math. Ser., 364, Longman, Harlow, 1997. Math. Review number not vailable.
  9. S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob--Meyer's type, Probab. Theory Related Fields 113 (1999), no. 4, 473--499. Math. Review 1717527.
  10. F. Pradeilles, Wavefront propagation for reaction-diffusion systems and backward SDEs, Ann. Probab. 26 (1998), no. 4, 1575--1613. Math. Review 2000e:35103.
  11. R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, N.J., 1970. Math. Review 43#445.

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