Probability Theory
A Note on FBSDE characterization of mean exit times
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 965-969.

In this Note, we present a new explicit characterization for a mean exit time problem recently treated by the author, in form of a quadratic Forward–Backward Stochastic Differential Equation (FBSDE) with a random terminal time. An a priori estimate and a uniqueness result for such a type of FBSDE are also proved, under certain conditions.

Dans cette Note on donne une nouvelle caractérisation explicite des temps de sortie moyens pour un probléme récemment introduit par l'auteur ; cette caractérisation est obtenue à partir d'une FBSDE quadratique à temps terminal aléatoire. On démontre aussi, sous certaines conditions, une estimation a priori, et un résultat d'unicité pour ce type d'équation différentielle stochastique directe et rétrograde.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.06.006
Makasu, Cloud 1

1 Department of Mathematics and Applied Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa
@article{CRMATH_2009__347_15-16_965_0,
     author = {Makasu, Cloud},
     title = {A {Note} on {FBSDE} characterization of mean exit times},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {965--969},
     publisher = {Elsevier},
     volume = {347},
     number = {15-16},
     year = {2009},
     doi = {10.1016/j.crma.2009.06.006},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.06.006/}
}
TY  - JOUR
AU  - Makasu, Cloud
TI  - A Note on FBSDE characterization of mean exit times
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 965
EP  - 969
VL  - 347
IS  - 15-16
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2009.06.006/
DO  - 10.1016/j.crma.2009.06.006
LA  - en
ID  - CRMATH_2009__347_15-16_965_0
ER  - 
%0 Journal Article
%A Makasu, Cloud
%T A Note on FBSDE characterization of mean exit times
%J Comptes Rendus. Mathématique
%D 2009
%P 965-969
%V 347
%N 15-16
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.06.006/
%R 10.1016/j.crma.2009.06.006
%G en
%F CRMATH_2009__347_15-16_965_0
Makasu, Cloud. A Note on FBSDE characterization of mean exit times. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 965-969. doi : 10.1016/j.crma.2009.06.006. http://www.numdam.org/articles/10.1016/j.crma.2009.06.006/

[1] Antonelli, F. Backward–forward stochastic differential equations, Ann. Appl. Probab., Volume 3 (1993), pp. 777-793

[2] Antonelli, F.; Hamadene, S. Existence of the solutions of backward–forward SDEs with continuous monotone coefficients, Statist. Probab. Lett., Volume 76 (2006), pp. 1559-1569

[3] Bahlali, K.; Essaky, E.H.; Pardoux, E. Existence, uniqueness and stability of backward stochastic differential equations with locally monotone coefficient, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 757-762

[4] Briand, P.; Hu, Y. BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields, Volume 136 (2006), pp. 604-618

[5] Darling, R.W.R.; Pardoux, E. Backwards stochastic differential equation with random terminal time and applications to semilinear elliptic partial differential equations, Ann. Probab., Volume 25 (1997), pp. 1135-1159

[6] El Karoui, N.; Peng, S.; Quenez, M.C. Backward stochastic differential equations in finance, Math. Finance, Volume 7 (1997), pp. 1-71

[7] El Karoui, N.; Rouge, R. Contingent claim pricing via utility maximization, Math. Finance, Volume 10 (2000) no. 2, pp. 259-276

[8] Hu, Y.; Imkeller, P.; Müller, M. Utility maximization in incomplete markets, Ann. Appl. Probab., Volume 15 (2005), pp. 1691-1712

[9] Hu, Y.; Peng, S. Solution of forward–backward stochastic differential equations, Probab. Theory Related Fields, Volume 103 (1995), pp. 273-283

[10] Kobylanski, M. Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., Volume 28 (2000), pp. 558-602

[11] Lepeltier, J.P.; San Martin, J. Backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett., Volume 32 (1997), pp. 425-430

[12] Lepeltier, J.P.; San Martin, J. Existence for BSDE with superlinear-quadratic coefficient, Stochastics Stochastics Rep., Volume 63 (1998), pp. 227-240

[13] Makasu, C. On mean exit time from a curvilinear domain, Statist. Probab. Lett., Volume 78 (2008), pp. 2859-2863

[14] Narita, K. No explosion criteria for stochastic differential equations, J. Math. Soc. Japan, Volume 34 (1982), pp. 191-203

[15] Pardoux, P.; Peng, S. Adapted solution of backward stochastic differential equation, Systems Control Lett., Volume 14 (1990), pp. 55-61

[16] Peng, S.; Shi, Y. Infinite horizon forward–backward stochastic differential equations, Stochastic Process. Appl., Volume 85 (2000), pp. 75-92

[17] Peng, S.; Wu, Z. Fully coupled forward–backward stochastic differential equations and applications to optimal control, Siam J. Control Optim., Volume 37 (1999), pp. 825-843

[18] Peng, S. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., Volume 32 (1991), pp. 61-74

[19] Rozkosz, A. On existence of solutions of BSDEs with continuous coefficient, Statist. Probab. Lett., Volume 67 (2004), pp. 249-256

[20] Sekine, J. On exponential hedging and related quadratic backward stochastic differential equations, Appl. Math. Optim., Volume 54 (2006), pp. 131-158

[21] Wu, Z.; Xu, M. Comparison theorems for forward backward SDEs, Statist. Probab. Lett., Volume 79 (2009), pp. 426-435

[22] Yin, J. On solutions of a class of infinite horizon FBSDEs, Statist. Probab. Lett., Volume 78 (2008), pp. 2412-2419

Cited by Sources:

Partial results of this Note were obtained when the author was holding a postdoc grant PRO12/1003 at the Mathematics Institute, University of Oslo, Norway.