Reflected backward stochastic differential equations with perturbations

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

Reflected backward stochastic differential equations with perturbations

Jasmina Djordjević ^, and Svetlana Janković

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 $η$.

Keywords: Reflected backward stochastic differential equations, perturbations, L^p-stability, L^p-closeness.

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

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, L^p 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, L^p 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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