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 is a sequence of scaled random walks or a sequence of martingales that converges to a brownian motion and if is a sequence of stopping times that converges to a stopping time , then the solution of the BSDE driven by with random terminal time converges to the solution of the BSDE driven by with random terminal time .
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}, mrnumber = {2218406}, zbl = {1185.60064}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps:2006006/} }
TY - JOUR AU - Toldo, Sandrine TI - Stability of solutions of BSDEs with random terminal time JO - ESAIM: Probability and Statistics PY - 2006 SP - 141 EP - 163 VL - 10 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2006006/ DO - 10.1051/ps:2006006 LA - en ID - PS_2006__10__141_0 ER -
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://archive.numdam.org/articles/10.1051/ps:2006006/
[1] Filtration stability of backward SDE's. Stochastic Anal. Appl. 18 (2000) 11-37. | Zbl
and ,[2] Convergence of Probability Measures, Second Edition. Wiley and Sons, New York (1999). | MR | Zbl
,[3] Donsker-type theorem for BSDEs. Electron. Comm. Probab. 6 (2001) 1-14 (electronic). | Zbl
, and ,[4] On the robustness of backward stochastic differential equations. Stochastic Process. Appl. 97 (2002) 229-253. | Zbl
, and ,[5] From Brownian motion to Schrödinger's equation, Springer-Verlag, Berlin Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 312 (1995). | Zbl
and ,[6] Stability in of martingales and backward equations under discretization of filtration. Stochastic Process. Appl. 75 (1998) 235-248. | Zbl
, and ,[7] Corrigendum to: “Stability in of martingales and backward equations under discretization of filtration”. Stochastic Process. Appl. 82 (1999) 335-338. | Zbl
, and ,[8] On weak convergence of filtrations. Séminaire de probabilités XXXV, Springer-Verlag, Berlin Heidelberg New York, Lect. Notes Math. 1755 (2001) 306-328. | Numdam | Zbl
, and ,[9] 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
and ,[10] Convergence in distribution and Skorokhod convergence for the general theory of processes. Probab. Theory Related Fields 89 (1991) 239-259. | Zbl
,[11] Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin Heidelberg New York (1987). | MR | Zbl
and ,[12] Brownian Motion and Stochastic Calculus, Second Edition. Springer-Verlag, Berlin Heidelberg New York (1991). | MR | Zbl
and ,[13] Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12 (2002) 302-316. | Zbl
, , and ,[14] Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37 (1991) 61-74. | Zbl
,[15] BSDEs with a random terminal time driven by a monotone generator and their links with PDEs. Stoch. Stoch. Rep. 76 (2004) 281-307. | Zbl
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