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$C^1$ weak Palis conjecture for nonsingular flows
Reflected backward stochastic differential equations with perturbations
Faculty of Science and Mathematics, University of Niš, Višegradska 33,18000 Niš, Serbia |
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 $η$.
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. |
show all references
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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