Stability of solutions of BSDEs with random terminal time
ESAIM: Probability and Statistics, Tome 10 (2006) , pp. 141-163.

In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time. More precisely, we are going to show that if $\left({W}^{n}\right)$ is a sequence of scaled random walks or a sequence of martingales that converges to a brownian motion $W$ and if $\left({\tau }^{n}\right)$ is a sequence of stopping times that converges to a stopping time $\tau$, then the solution of the BSDE driven by ${W}^{n}$ with random terminal time ${\tau }^{n}$ converges to the solution of the BSDE driven by $W$ with random terminal time $\tau$.

DOI : https://doi.org/10.1051/ps:2006006
Classification : 60H10,  60Fxx,  60G40
Mots clés : backward stochastic differential equations (BSDE), stability of BSDEs, weak convergence of filtrations, stopping times
@article{PS_2006__10__141_0,
author = {Toldo, Sandrine},
title = {Stability of solutions of BSDEs with random terminal time},
journal = {ESAIM: Probability and Statistics},
pages = {141--163},
publisher = {EDP-Sciences},
volume = {10},
year = {2006},
doi = {10.1051/ps:2006006},
zbl = {1185.60064},
mrnumber = {2218406},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2006006/}
}
Toldo, Sandrine. Stability of solutions of BSDEs with random terminal time. ESAIM: Probability and Statistics, Tome 10 (2006) , pp. 141-163. doi : 10.1051/ps:2006006. http://www.numdam.org/articles/10.1051/ps:2006006/

[1] F. Antonelli and A. Kohatsu-Higa, Filtration stability of backward SDE's. Stochastic Anal. Appl. 18 (2000) 11-37. | Zbl 0953.60044

[2] P. Billingsley, Convergence of Probability Measures, Second Edition. Wiley and Sons, New York (1999). | MR 1700749 | Zbl 0172.21201

[3] P. Briand, B. Delyon and J. Mémin, Donsker-type theorem for BSDEs. Electron. Comm. Probab. 6 (2001) 1-14 (electronic). | Zbl 0977.60067

[4] P. Briand, B. Delyon and J. Mémin, On the robustness of backward stochastic differential equations. Stochastic Process. Appl. 97 (2002) 229-253. | Zbl 1058.60041

[5] K.L. Chung and Z.X. Zhao, From Brownian motion to Schrödinger's equation, Springer-Verlag, Berlin Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 312 (1995). | Zbl 0819.60068

[6] F. Coquet, V. Mackevičius and J. Mémin, Stability in $𝐃$ of martingales and backward equations under discretization of filtration. Stochastic Process. Appl. 75 (1998) 235-248. | Zbl 0932.60047

[7] F. Coquet, V. Mackevičius and J. Mémin, Corrigendum to: “Stability in $𝐃$ of martingales and backward equations under discretization of filtration”. Stochastic Process. Appl. 82 (1999) 335-338. | Zbl 0997.60061

[8] F. Coquet, J. Mémin and L. Słomiński, On weak convergence of filtrations. Séminaire de probabilités XXXV, Springer-Verlag, Berlin Heidelberg New York, Lect. Notes Math. 1755 (2001) 306-328. | Numdam | Zbl 0987.60009

[9] J. Haezendonck and F. Delbaen, Caractérisation de la tribu des événements antérieurs à un temps d'arrêt pour un processus stochastique. Acad. Roy. Belg., Bulletin de la Classe Scientifique 56 (1970) 1085-1092. | Zbl 0232.60032

[10] D.N. Hoover, Convergence in distribution and Skorokhod convergence for the general theory of processes. Probab. Theory Related Fields 89 (1991) 239-259. | Zbl 0725.60005

[11] J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin Heidelberg New York (1987). | MR 959133 | Zbl 0635.60021

[12] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Second Edition. Springer-Verlag, Berlin Heidelberg New York (1991). | MR 1121940 | Zbl 0734.60060

[13] J. Ma, P. Protter, J. San Martín and S. Torres, Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12 (2002) 302-316. | Zbl 1017.60074

[14] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37 (1991) 61-74. | Zbl 0739.60060

[15] M. Royer, BSDEs with a random terminal time driven by a monotone generator and their links with PDEs. Stoch. Stoch. Rep. 76 (2004) 281-307. | Zbl 1055.60062