# American Institute of Mathematical Sciences

April  2018, 38(4): 1833-1848. doi: 10.3934/dcds.2018075

## Reflected backward stochastic differential equations with perturbations

 Faculty of Science and Mathematics, University of Niš, Višegradska 33,18000 Niš, Serbia

* Corresponding author

Received  October 2016 Revised  October 2017 Published  January 2018

Fund Project: Supported by Grant No 174007 of MNTRS.

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 $η$.

Citation: Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
##### References:
 [1] A. Aman, $L_p$-solutions of reflected generalized backward stochastic differential equations with non-Lipschitz coefficients, Random Operators/Stochastic. Eqs., 17 (2009), 201-219. [2] A. Aman, $L_p$-solutions of generalized backward stochastic differential equations with barrier, Afr. Diaspora J. Math, 8 (2009), 68-80. [3] 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. [4] 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. [5] D. Bainov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Netherlands, 1992. [6] 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. [7] 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. [8] N. El-Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. [9] M. I. Friedlin, A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, Berlin, 1984. [10] 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. [11] 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. [12] S. Hamadène and J. P. Lepeltier, Backward equations, stochastic control and zero-sum stochastic differential games, Stoch. Stoch. Rep., 54 (1995), 221-231. [13] S. Hamadène, Reflected BSDEs with discontinuous barrier and applications, Stoch. Stoch. Rep., 74 (2002), 571-596. [14] S. Hamadène and Y. Ouknine, Reflected backward stochastic differential equations with jumps and random obstacle, Electron. J. of Probab., 8 (2003), 1-20. [15] S. Hamadène and A. Popier, Lp-solutions for Reflected Backward Stochastic Differential Equations, Stochastics and Dynamics, 12 (2012), 1150016, 35 pp. [16] S. Hamadène and M. Jeanblanc, On the stopping and starting problem: Application to reversible investment, Math. Oper. Res., 32 (2007), 182-192. [17] S. Jankovic, M. Jovanovic and J. Djordjevic, Perturbed backward stochastic differential equations, Math. Comput. Modelling, 55 (2012), 1734-1745. [18] R. Khasminskii, On stochastic processes deffined by differential equations with a small parameter, Theory Probab. Appl., 11 (1966), 240-259. [19] 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. [20] 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. [21] X. Mao, Stochastic Differential Equations and Applications, second edition, Horvood, Chichester, UK, 2008. [22] A. Matoussi, Reflected solutions of backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett., 34 (1997), 347-354. [23] Y. Ouknine, Reflected BSDE with jumps, Stoch. Stoch. Rep., 65 (1998), 111-125. [24] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Letters, 14 (1990), 55-61. [25] 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. [26] É Pardoux and A. Rascanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl., 76 (1998), 191-215. [27] 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. [28] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Rep., 37 (1991), 61-74. [29] 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. [30] Y. Ren and L. Hu, Reflected backward stochastic differential equations driven by Lévy processes, Statist. Probab. Lett., 77 (2007), 1559-1566. [31] 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. [32] 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. [33] 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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##### References:
 [1] A. Aman, $L_p$-solutions of reflected generalized backward stochastic differential equations with non-Lipschitz coefficients, Random Operators/Stochastic. Eqs., 17 (2009), 201-219. [2] A. Aman, $L_p$-solutions of generalized backward stochastic differential equations with barrier, Afr. Diaspora J. Math, 8 (2009), 68-80. [3] 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. [4] 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. [5] D. Bainov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Netherlands, 1992. [6] 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. [7] 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. [8] N. El-Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. [9] M. I. Friedlin, A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, Berlin, 1984. [10] 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. [11] 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. [12] S. Hamadène and J. P. Lepeltier, Backward equations, stochastic control and zero-sum stochastic differential games, Stoch. Stoch. Rep., 54 (1995), 221-231. [13] S. Hamadène, Reflected BSDEs with discontinuous barrier and applications, Stoch. Stoch. Rep., 74 (2002), 571-596. [14] S. Hamadène and Y. Ouknine, Reflected backward stochastic differential equations with jumps and random obstacle, Electron. J. of Probab., 8 (2003), 1-20. [15] S. Hamadène and A. Popier, Lp-solutions for Reflected Backward Stochastic Differential Equations, Stochastics and Dynamics, 12 (2012), 1150016, 35 pp. [16] S. Hamadène and M. Jeanblanc, On the stopping and starting problem: Application to reversible investment, Math. Oper. Res., 32 (2007), 182-192. [17] S. Jankovic, M. Jovanovic and J. Djordjevic, Perturbed backward stochastic differential equations, Math. Comput. Modelling, 55 (2012), 1734-1745. [18] R. Khasminskii, On stochastic processes deffined by differential equations with a small parameter, Theory Probab. Appl., 11 (1966), 240-259. [19] 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. [20] 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. [21] X. Mao, Stochastic Differential Equations and Applications, second edition, Horvood, Chichester, UK, 2008. [22] A. Matoussi, Reflected solutions of backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett., 34 (1997), 347-354. [23] Y. Ouknine, Reflected BSDE with jumps, Stoch. Stoch. Rep., 65 (1998), 111-125. [24] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Letters, 14 (1990), 55-61. [25] 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. [26] É Pardoux and A. Rascanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl., 76 (1998), 191-215. [27] 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. [28] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Rep., 37 (1991), 61-74. [29] 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. [30] Y. Ren and L. Hu, Reflected backward stochastic differential equations driven by Lévy processes, Statist. Probab. Lett., 77 (2007), 1559-1566. [31] 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. [32] 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. [33] 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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