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Reflected backward stochastic differential equations with perturbations

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Supported by Grant No 174007 of MNTRS

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  • This paper deals with a large class of reflected backward stochastic differential equations whose generators arbitrarily depend on a small parameter. The solutions of these equations, named the perturbed equations, are compared in the $L^p$-sense, $p∈ ]1,2[$, with the solutions of the appropriate equations of the equal type, independent of a small parameter and named the unperturbed equations. Conditions under which the solution of the unperturbed equation is $L^p$-stable are given. It is shown that for an arbitrary $η>0$ there exists an interval $[t(η), T]\subset [0,T]$ on which the $L^p$-difference between the solutions of both the perturbed and unperturbed equations is less than $η$.

    Mathematics Subject Classification: Primary: 60H35, 93E10; Secondary: 93E25.

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  •   A. Aman , $ L_p$-solutions of reflected generalized backward stochastic differential equations with non-Lipschitz coefficients, Random Operators/Stochastic. Eqs., 17 (2009) , 201-219. 
      A. Aman , $L_p$-solutions of generalized backward stochastic differential equations with barrier, Afr. Diaspora J. Math, 8 (2009) , 68-80. 
      K. Bahlali , El. Essaky  and  Y. Ouknine , Reflected backward stochastic differential equations with jumps and locally Lipschitz coefficient, Random Oper. Stochastic Equations, 10 (2002) , 335-350. 
      K. Bahlali , El. Essaky  and  Y. Ouknine , Reflected backward stochastic differential equations with jumps and locally monotone coefficient, Stoch. Anal. Appl., 22 (2004) , 939-970. 
      D. Bainov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Netherlands, 1992.
      B. El-Asri  and  S. Hamadène , The finite horizon optimal multi-modes switching problem: The viscosity solution approach, Appl. Math. Optim., 60 (2009) , 213-235. 
      N. El-Karoui , C. Kapoudjian , E. Pardoux , S. Peng  and  M.-C. Quenez , Reflected solutions of backward SDE s, and related obstacle problems for PDE s, Ann. Probab., 25 (1997) , 702-737. 
      N. El-Karoui , S. Peng  and  M. C. Quenez , Backward stochastic differential equations in finance, Math. Finance, 7 (1997) , 1-71. 
      M. I. Friedlin, A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, Berlin, 1984.
      A. Gégout-Petit, A Filtrage d'un processus partiellement observé et équations differentielles stochastiques rétrogrades réfléchies, Thése de doctorat l'Université de Provence-Aix-Marseille, 1995.
      S. Hamadène , BSDEs and risk sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stochastic Process. Appl., 107 (2003) , 145-169. 
      S. Hamadène  and  J. P. Lepeltier , Backward equations, stochastic control and zero-sum stochastic differential games, Stoch. Stoch. Rep., 54 (1995) , 221-231. 
      S. Hamadène , Reflected BSDEs with discontinuous barrier and applications, Stoch. Stoch. Rep., 74 (2002) , 571-596. 
      S. Hamadène  and  Y. Ouknine , Reflected backward stochastic differential equations with jumps and random obstacle, Electron. J. of Probab., 8 (2003) , 1-20. 
      S. Hamadène and A. Popier, Lp-solutions for Reflected Backward Stochastic Differential Equations, Stochastics and Dynamics, 12 (2012), 1150016, 35 pp.
      S. Hamadène  and  M. Jeanblanc , On the stopping and starting problem: Application to reversible investment, Math. Oper. Res., 32 (2007) , 182-192. 
      S. Jankovic , M. Jovanovic  and  J. Djordjevic , Perturbed backward stochastic differential equations, Math. Comput. Modelling, 55 (2012) , 1734-1745. 
      R. Khasminskii , On stochastic processes deffined by differential equations with a small parameter, Theory Probab. Appl., 11 (1966) , 240-259. 
      J. P. Lepeltier , A. Matoussi  and  M. Xu , Reflected backward stochastic differential equations under monotonicity and general increasing growth conditions, Adv. Appl. Probab., 37 (2005) , 134-159. 
      J. P. Lepeltier  and  M. Xu , Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier, Statist. Probab. Lett., 75 (2005) , 58-66. 
      X. Mao, Stochastic Differential Equations and Applications, second edition, Horvood, Chichester, UK, 2008.
      A. Matoussi , Reflected solutions of backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett., 34 (1997) , 347-354. 
      Y. Ouknine , Reflected BSDE with jumps, Stoch. Stoch. Rep., 65 (1998) , 111-125. 
      E. Pardoux  and  S. G. Peng , Adapted solution of a backward stochastic differential equation, Systems Control Letters, 14 (1990) , 55-61. 
      E. Pardoux and S. G. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in: Stochastic Partial Differential Equations and Their Applications, (Charlotte, NC, 1991) (B. Rozowskii and R. Sowers, eds. ), Lecture Notes in Control and Information Sci., Springer, Berlin, 176 (1992), 200-217.
      É Pardoux  and  A. Rascanu , Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl., 76 (1998) , 191-215. 
      É Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, in: Nonlinear analysis, differential equations and control (Montreal, QC, 1998), Volume 528 of NATO Sci. Ser. C Math. Phys. Sci. (Kluwer Academic Publishers, Dordrecht, (1999), 503-549.
      S. Peng , Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Rep., 37 (1991) , 61-74. 
      Y. Ren  and  N. Xia , Generalized reflected BSDEs and an obstacle problem for PDEs with a nonlinear Neumann boundary condition, Stoch. Anal. Appl., 24 (2006) , 1013-1033. 
      Y. Ren  and  L. Hu , Reflected backward stochastic differential equations driven by Lévy processes, Statist. Probab. Lett., 77 (2007) , 1559-1566. 
      A. Roskosz and L. Slominski, Lp solutions of reflected BSDEs under monotonicity condition, Stochastic Process. Appl., 122 (2012), 3875-3900, arXiv: 1205.6737. doi: 10.1016/j.spa.2012.07.006.
      J. Stoyanov, Regularly perturbed stochastic differential systems with an internal random noise, in: Proc. 2ndWorld Congress Nonlin. Anal., Nonlinear Anal., 30 (1997), 4105-4111. doi: d10.1016/S0362-546X(97)00158-2oi.
      J. Stoyanov  and  D. Botev , Quantitative results for perturbed stochastic differential equations, J. Appl. Math. Stoch. Anal., 9 (1996) , 255-261.  doi: 10.1155/S104895339600024X.
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