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.
Accepted:
Published online:
Cloud Makasu 1
@article{CRMATH_2009__347_15-16_965_0, author = {Cloud Makasu}, 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}, }
Cloud Makasu. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.006/
[1] Backward–forward stochastic differential equations, Ann. Appl. Probab., Volume 3 (1993), pp. 777-793
[2] Existence of the solutions of backward–forward SDEs with continuous monotone coefficients, Statist. Probab. Lett., Volume 76 (2006), pp. 1559-1569
[3] 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] BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields, Volume 136 (2006), pp. 604-618
[5] 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] Backward stochastic differential equations in finance, Math. Finance, Volume 7 (1997), pp. 1-71
[7] Contingent claim pricing via utility maximization, Math. Finance, Volume 10 (2000) no. 2, pp. 259-276
[8] Utility maximization in incomplete markets, Ann. Appl. Probab., Volume 15 (2005), pp. 1691-1712
[9] Solution of forward–backward stochastic differential equations, Probab. Theory Related Fields, Volume 103 (1995), pp. 273-283
[10] Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., Volume 28 (2000), pp. 558-602
[11] Backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett., Volume 32 (1997), pp. 425-430
[12] Existence for BSDE with superlinear-quadratic coefficient, Stochastics Stochastics Rep., Volume 63 (1998), pp. 227-240
[13] On mean exit time from a curvilinear domain, Statist. Probab. Lett., Volume 78 (2008), pp. 2859-2863
[14] No explosion criteria for stochastic differential equations, J. Math. Soc. Japan, Volume 34 (1982), pp. 191-203
[15] Adapted solution of backward stochastic differential equation, Systems Control Lett., Volume 14 (1990), pp. 55-61
[16] Infinite horizon forward–backward stochastic differential equations, Stochastic Process. Appl., Volume 85 (2000), pp. 75-92
[17] Fully coupled forward–backward stochastic differential equations and applications to optimal control, Siam J. Control Optim., Volume 37 (1999), pp. 825-843
[18] Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., Volume 32 (1991), pp. 61-74
[19] On existence of solutions of BSDEs with continuous coefficient, Statist. Probab. Lett., Volume 67 (2004), pp. 249-256
[20] On exponential hedging and related quadratic backward stochastic differential equations, Appl. Math. Optim., Volume 54 (2006), pp. 131-158
[21] Comparison theorems for forward backward SDEs, Statist. Probab. Lett., Volume 79 (2009), pp. 426-435
[22] 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.
Comments - Politique