# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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, (1984). Google Scholar [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. doi: 10.1007/s10255-011-0068-8. Google Scholar [3] R. Carmona and M. Ludkovski, Pricing asset scheduling flexibility using optimal switching,, Appl. Math. Finance, 15 (2008), 405. doi: 10.1080/13504860802170507. Google Scholar [4] J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games,, Ann. Probab., 24 (1996), 2024. doi: 10.1214/aop/1041903216. Google Scholar [5] P. Dupuis and H. Ishii, SDEs with oblique reflection on nonsmooth domains,, Ann. Probab., 21 (1993), 554. doi: 10.1214/aop/1176989415. Google Scholar [6] N. El Karoui, Les aspects probabilistes du contrôle stochastique,, Ninth Saint Flour Probability Summer School - 1979 (Saint Flour, (1979), 73. Google Scholar [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. doi: 10.1214/aop/1024404416. Google Scholar [8] A. Gegout-Petit and E. Pardoux, Equations différentielles stochastiques rétrogrades réfléchies dans un convexe,, Stochastics Stochastic Rep., 57 (1996), 111. doi: 10.1080/17442509608834054. Google Scholar [9] S. Hamadène and M. Jeanblanc, On the starting and stopping problem: Application in reversible investments,, Math. Oper. Res., 32 (2007), 182. doi: 10.1287/moor.1060.0228. Google Scholar [10] Y. Hu and S. Peng, On the comparison theorem for multi-dimensional BSDEs,, C. R. Math. Acad. Sci. Paris, 343 (2006), 135. doi: 10.1016/j.crma.2006.05.019. Google Scholar [11] Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching,, Probab. Theory Related Fields, 147 (2010), 89. doi: 10.1007/s00440-009-0202-1. Google Scholar [12] P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions,, Comm. Pure Appl. Math., 37 (1984), 511. doi: 10.1002/cpa.3160370408. Google Scholar [13] P. A. Meyer, Un cours sur les intégrales stochastiques., Séminaire de Probabilités, (1976), 245. Google Scholar [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. doi: 10.1016/j.anihpb.2004.12.002. Google Scholar [15] H. Pham, V. Ly Vath and X. Y. Zhou, Optimal switching over multiple regimes,, SIAM J. Control Optim., 48 (2009), 2217. doi: 10.1137/070709372. Google Scholar [16] S. Ramasubramanian, Reflected backward stochastic differential equations in an orthant,, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 347. doi: 10.1007/BF02829759. Google Scholar [17] S. Tang and S. Hou, Switching games of stochastic differential systems,, SIAM J. Control Optim., 46 (2007), 900. doi: 10.1137/050642204. Google Scholar [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. doi: 10.1080/17442509308833860. Google Scholar [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. doi: 10.1137/080738349. Google Scholar

show all references

##### References:
 [1] A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities,, Gauthier-Villars, (1984). Google Scholar [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. doi: 10.1007/s10255-011-0068-8. Google Scholar [3] R. Carmona and M. Ludkovski, Pricing asset scheduling flexibility using optimal switching,, Appl. Math. Finance, 15 (2008), 405. doi: 10.1080/13504860802170507. Google Scholar [4] J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games,, Ann. Probab., 24 (1996), 2024. doi: 10.1214/aop/1041903216. Google Scholar [5] P. Dupuis and H. Ishii, SDEs with oblique reflection on nonsmooth domains,, Ann. Probab., 21 (1993), 554. doi: 10.1214/aop/1176989415. Google Scholar [6] N. El Karoui, Les aspects probabilistes du contrôle stochastique,, Ninth Saint Flour Probability Summer School - 1979 (Saint Flour, (1979), 73. Google Scholar [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. doi: 10.1214/aop/1024404416. Google Scholar [8] A. Gegout-Petit and E. Pardoux, Equations différentielles stochastiques rétrogrades réfléchies dans un convexe,, Stochastics Stochastic Rep., 57 (1996), 111. doi: 10.1080/17442509608834054. Google Scholar [9] S. Hamadène and M. Jeanblanc, On the starting and stopping problem: Application in reversible investments,, Math. Oper. Res., 32 (2007), 182. doi: 10.1287/moor.1060.0228. Google Scholar [10] Y. Hu and S. Peng, On the comparison theorem for multi-dimensional BSDEs,, C. R. Math. Acad. Sci. Paris, 343 (2006), 135. doi: 10.1016/j.crma.2006.05.019. Google Scholar [11] Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching,, Probab. Theory Related Fields, 147 (2010), 89. doi: 10.1007/s00440-009-0202-1. Google Scholar [12] P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions,, Comm. Pure Appl. Math., 37 (1984), 511. doi: 10.1002/cpa.3160370408. Google Scholar [13] P. A. Meyer, Un cours sur les intégrales stochastiques., Séminaire de Probabilités, (1976), 245. Google Scholar [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. doi: 10.1016/j.anihpb.2004.12.002. Google Scholar [15] H. Pham, V. Ly Vath and X. Y. Zhou, Optimal switching over multiple regimes,, SIAM J. Control Optim., 48 (2009), 2217. doi: 10.1137/070709372. Google Scholar [16] S. Ramasubramanian, Reflected backward stochastic differential equations in an orthant,, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 347. doi: 10.1007/BF02829759. Google Scholar [17] S. Tang and S. Hou, Switching games of stochastic differential systems,, SIAM J. Control Optim., 46 (2007), 900. doi: 10.1137/050642204. Google Scholar [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. doi: 10.1080/17442509308833860. Google Scholar [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. doi: 10.1137/080738349. Google Scholar
 [1] Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028 [2] Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013 [3] Yaozhong Hu, David Nualart, Xiaobin Sun, Yingchao Xie. Smoothness of density for stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3615-3631. doi: 10.3934/dcdsb.2018307 [4] Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075 [5] Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803 [6] Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002 [7] Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115 [8] Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285 [9] Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565 [10] Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013 [11] Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905 [12] Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585 [13] Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051 [14] Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297 [15] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [16] Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501 [17] Boling Guo, Guoli Zhou. On the backward uniqueness of the stochastic primitive equations with additive noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3157-3174. doi: 10.3934/dcdsb.2018305 [18] Xiao-Qian Jiang, Lun-Chuan Zhang. A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-978. doi: 10.3934/dcdss.2019065 [19] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929 [20] Qi Lü, Xu Zhang. Transposition method for backward stochastic evolution equations revisited, and its application. Mathematical Control & Related Fields, 2015, 5 (3) : 529-555. doi: 10.3934/mcrf.2015.5.529

2018 Impact Factor: 1.143