November  2015, 35(11): 5447-5465. doi: 10.3934/dcds.2015.35.5447

Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations

1. 

IRMAR, Université Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex

2. 

School of Mathematical Science, Fudan University, Shanghai 200433

Received  November 2013 Revised  November 2014 Published  May 2015

This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates.
Citation: Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447
References:
[1]

A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities, Gauthier-Villars, Montrouge, 1984.

[2]

R. Buckdahn and J. Li, Stochastic differential games with reflection and related obstacle problems for Isaacs equations, Acta Math. Appl. Sin. Engl. Ser., 27 (2011), 647-678. doi: 10.1007/s10255-011-0068-8.

[3]

R. Carmona and M. Ludkovski, Pricing asset scheduling flexibility using optimal switching, Appl. Math. Finance, 15 (2008), 405-447. doi: 10.1080/13504860802170507.

[4]

J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24 (1996), 2024-2056. doi: 10.1214/aop/1041903216.

[5]

P. Dupuis and H. Ishii, SDEs with oblique reflection on nonsmooth domains, Ann. Probab., 21 (1993), 554-580. doi: 10.1214/aop/1176989415.

[6]

N. El Karoui, Les aspects probabilistes du contrôle stochastique, Ninth Saint Flour Probability Summer School - 1979 (Saint Flour, 1979), Lecture Notes in Math., 876, Springer, Berlin, 1981, 73-238.

[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. doi: 10.1214/aop/1024404416.

[8]

A. Gegout-Petit and E. Pardoux, Equations différentielles stochastiques rétrogrades réfléchies dans un convexe, Stochastics Stochastic Rep., 57 (1996), 111-128. doi: 10.1080/17442509608834054.

[9]

S. Hamadène and M. Jeanblanc, On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), 182-192. doi: 10.1287/moor.1060.0228.

[10]

Y. Hu and S. Peng, On the comparison theorem for multi-dimensional BSDEs, C. R. Math. Acad. Sci. Paris, 343 (2006), 135-140. doi: 10.1016/j.crma.2006.05.019.

[11]

Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147 (2010), 89-121. doi: 10.1007/s00440-009-0202-1.

[12]

P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537. doi: 10.1002/cpa.3160370408.

[13]

P. A. Meyer, Un cours sur les intégrales stochastiques. Séminaire de Probabilités, X, Lecture Notes in Math., 511, Springer, Berlin, 1976, 245-400.

[14]

S. Peng and M. Xu, The smallest $g$-supermartingale and reflected BSDE with single and double $L^2$ obstacles, Ann. Inst. H. Poincaré Probab. Statist., 41 (2005), 605-630. doi: 10.1016/j.anihpb.2004.12.002.

[15]

H. Pham, V. Ly Vath and X. Y. Zhou, Optimal switching over multiple regimes, SIAM J. Control Optim., 48 (2009), 2217-2253. doi: 10.1137/070709372.

[16]

S. Ramasubramanian, Reflected backward stochastic differential equations in an orthant, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 347-360. doi: 10.1007/BF02829759.

[17]

S. Tang and S. Hou, Switching games of stochastic differential systems, SIAM J. Control Optim., 46 (2007), 900-929. doi: 10.1137/050642204.

[18]

S. Tang and J. Yong, Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach, Stochastics Stochastics Rep., 45 (1993), 145-176. doi: 10.1080/17442509308833860.

[19]

S. Tang, W. Zhong and H. Koo, Optimal switching of one-dimensional reflected BSDEs and associated multidimensional BSDEs with oblique reflection, SIAM J. Control Optim., 49 (2011), 2279-2317. doi: 10.1137/080738349.

show all references

References:
[1]

A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities, Gauthier-Villars, Montrouge, 1984.

[2]

R. Buckdahn and J. Li, Stochastic differential games with reflection and related obstacle problems for Isaacs equations, Acta Math. Appl. Sin. Engl. Ser., 27 (2011), 647-678. doi: 10.1007/s10255-011-0068-8.

[3]

R. Carmona and M. Ludkovski, Pricing asset scheduling flexibility using optimal switching, Appl. Math. Finance, 15 (2008), 405-447. doi: 10.1080/13504860802170507.

[4]

J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24 (1996), 2024-2056. doi: 10.1214/aop/1041903216.

[5]

P. Dupuis and H. Ishii, SDEs with oblique reflection on nonsmooth domains, Ann. Probab., 21 (1993), 554-580. doi: 10.1214/aop/1176989415.

[6]

N. El Karoui, Les aspects probabilistes du contrôle stochastique, Ninth Saint Flour Probability Summer School - 1979 (Saint Flour, 1979), Lecture Notes in Math., 876, Springer, Berlin, 1981, 73-238.

[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. doi: 10.1214/aop/1024404416.

[8]

A. Gegout-Petit and E. Pardoux, Equations différentielles stochastiques rétrogrades réfléchies dans un convexe, Stochastics Stochastic Rep., 57 (1996), 111-128. doi: 10.1080/17442509608834054.

[9]

S. Hamadène and M. Jeanblanc, On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), 182-192. doi: 10.1287/moor.1060.0228.

[10]

Y. Hu and S. Peng, On the comparison theorem for multi-dimensional BSDEs, C. R. Math. Acad. Sci. Paris, 343 (2006), 135-140. doi: 10.1016/j.crma.2006.05.019.

[11]

Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147 (2010), 89-121. doi: 10.1007/s00440-009-0202-1.

[12]

P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537. doi: 10.1002/cpa.3160370408.

[13]

P. A. Meyer, Un cours sur les intégrales stochastiques. Séminaire de Probabilités, X, Lecture Notes in Math., 511, Springer, Berlin, 1976, 245-400.

[14]

S. Peng and M. Xu, The smallest $g$-supermartingale and reflected BSDE with single and double $L^2$ obstacles, Ann. Inst. H. Poincaré Probab. Statist., 41 (2005), 605-630. doi: 10.1016/j.anihpb.2004.12.002.

[15]

H. Pham, V. Ly Vath and X. Y. Zhou, Optimal switching over multiple regimes, SIAM J. Control Optim., 48 (2009), 2217-2253. doi: 10.1137/070709372.

[16]

S. Ramasubramanian, Reflected backward stochastic differential equations in an orthant, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 347-360. doi: 10.1007/BF02829759.

[17]

S. Tang and S. Hou, Switching games of stochastic differential systems, SIAM J. Control Optim., 46 (2007), 900-929. doi: 10.1137/050642204.

[18]

S. Tang and J. Yong, Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach, Stochastics Stochastics Rep., 45 (1993), 145-176. doi: 10.1080/17442509308833860.

[19]

S. Tang, W. Zhong and H. Koo, Optimal switching of one-dimensional reflected BSDEs and associated multidimensional BSDEs with oblique reflection, SIAM J. Control Optim., 49 (2011), 2279-2317. doi: 10.1137/080738349.

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