Geodesic
A backward stochastic differential equation without strong solution
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 2, pp. 390-396 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In [R. Buckdahn, H.-J. Engelbert, and A. Răşcanu, Theory Probab. Appl., 49 (2005), pp. 16–50] the notion of a weak solution of a general backward stochastic differential equation (BSDE) was introduced. There was also given an example of a weak solution for a certain BSDE which is not a strong solution, i.e., not a solution in the classical sense. However, the solution of the BSDE which was considered is not unique in law and, as was pointed out, there exist also strong solutions of this BSDE. In the present paper, we will remove this insufficiency and give an example of a BSDE which has a weak solution but does not possess any strong solution.
Keywords: backward stochastic differential equations, weak solutions, strong solutions, Tsirelson example. 
@article{TVP_2005_50_2_a13,
     author = {R. Buckdahn and H. J. Engelbert},
     title = {A backward stochastic differential equation without strong solution},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {390--396},
     year = {2005},
     volume = {50},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a13/}
}
TY  - JOUR
AU  - R. Buckdahn
AU  - H. J. Engelbert
TI  - A backward stochastic differential equation without strong solution
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2005
SP  - 390
EP  - 396
VL  - 50
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a13/
LA  - en
ID  - TVP_2005_50_2_a13
ER  - 
%0 Journal Article
%A R. Buckdahn
%A H. J. Engelbert
%T A backward stochastic differential equation without strong solution
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2005
%P 390-396
%V 50
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a13/
%G en
%F TVP_2005_50_2_a13
R. Buckdahn; H. J. Engelbert. A backward stochastic differential equation without strong solution. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 2, pp. 390-396. http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a13/

[1] Antonelli F., Ma J., “Weak solutions of forward-backward SDE 's”, Stochastic Anal. Appl., 21:3 (2003), 493–514 | DOI | MR | Zbl

[2] Bahlali K., Essaky E. H., Hassani M., Pardoux E., “Existence, uniqueness and stability of backward stochastic differential equations with locally monotone coefficient”, C. R. Math. Acad. Sci. Paris, 335:9 (2002), 757–762 | MR | Zbl

[3] Buckdahn R., Engelbert H.-J., Răşcanu A., “On weak solutions of backward stochastic differential equations”, Teoriya veroyatn. i ee primen., 49:1 (2004), 70–108 | MR | Zbl

[4] El Karoui N., Mazliak L. (eds.), Backward Stochastic Differential Equations, Longman, Harlow, 1997, 221 pp. | MR

[5] Hamadène S., Lepeltier J.-P., Peng S., “BSDE s with continuous coefficients and stochastic differential games”, Backward Stochastic Differential Equations, eds. N. El Karoui and Z. Mazliak, Longman, Harlow, 1997, 115–128 | MR | Zbl

[6] Lepeltier J.-P., San Martin J., “Backward stochastic differential equations with continuous coefficient”, Statist. Probab. Lett., 32:4 (1997), 425–430 | DOI | MR | Zbl

[7] Meyer P. A., Zheng W. A., “Tightness criteria for laws of semimartingales”, Ann. Inst. H. Poincaré. Probab. Statist., 20:4 (1984), 353–372 | MR | Zbl

[8] Pardoux E., Peng S., “Adapted solution of a backward stochastic differential equation”, Systems Control Lett., 14:1 (1990), 55–61 | DOI | MR | Zbl

[9] Revuz D., Yor M., Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1991, 533 pp. | MR | Zbl

[10] Tsirelson B. S., “Odin primer stokhasticheskogo differentsialnogo uravneniya, ne imeyuschego silnogo resheniya”, Teoriya veroyatn. i ee primen., 20:2 (1975), 427–430 | MR

[11] Watanabe S., Yamada T., “On the uniqueness of solutions of stochastic differential equations”, J. Math. Kyoto Univ., 11 (1971), 155–167 | MR | Zbl