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The Cauchy problem of Backward Stochastic SuperParabolic Equations with Quadratic Growth
1. School of Mathematical Sciences, Fudan University, Shanghai 200433, China; 
2. Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, and Key Laboratory of Mathematics for Nonlinear Sciences(Fudan University), Ministry of Education, Shanghai 200433, China 
References:
[1] 
Bismut, J.M:Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384404(1973), 
[2] 
Bismut, J.M:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control. Optim. 14, 419444 (1976), 
[3] 
Briand, P.H, Hu, Y:BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Fields. 136, 604618 (2006), 
[4] 
Briand, P.H, Hu, Y:Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields. 141, 543567 (2008), 
[5] 
Chen, S.:Introduction to Modern Partial Differential Equations. Science Press, Beijing (2005), 
[6] 
Du, K.:Backward Stochastic Differential Equations and Their Applications. Dissertation, School of Mathematical Sciences, Fudan University (2011), 
[7] 
Du, K, Chen, S:Backward stochastic partial differential equations with quadratic growth. J. Math. Anal. Appl. 419, 447468 (2012), 
[8] 
Du, K, Qiu, J, Tang, S:L^{p} theory for superparabolic backward stochastic partial differential equations in the whole space. Appl. Math. Optim. 65, 175219 (2012), 
[9] 
Du, K, Tang, S:Strong solution of backward stochastic partial differential equations in C^{2} domains. Probab. Theory Related Fields. 154, 255285 (2010), 
[10] 
Du, K, Zhang, Q:Semilinear degenerate backward stochastic partial differential equations and associated forward backward stochastic differential equations. Stoch. Proc. Appl. 123, 16161637 (2013), 
[11] 
El Karoui, N, Hamadène, S:BSDEs and risksensitive control, zerosum and nonzerosum game problems of stochastic functional differential equations. Stochast. Process. Appl. 107, 145169 (2003), 
[12] 
Frei, C, dos Reis, G:A financial market with interacting investors:does an equilibrium exist? Math. Finan. Econ. 4, 161182 (2011), 
[13] 
Fujita, H:On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations. Proc. Sympos. Pure Math. 18, 105113 (1969), 
[14] 
Hu, Y, Tang, S:Multidimensional backward stochastic differential equations of diagonally quadratic generators. Stoch. Proc. Appl. 126, 10661086 (2016), 
[15] 
Kobylanski, M:Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558602 (2000), 
[16] 
Lepeltier, J.P, San Martin, J:Backward stochastic differential equations with continuous coefficients. Statist. Probab. Lett. 32, 425430 (1997), 
[17] 
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 5561 (1990), 
[18] 
Qiu, J, Tang, S:Maximum principle for quasilinear backward stochastic partial differential equations. J. Funct. Anal. 262, 24362480 (2012), 
[19] 
Tang, S:General linear quadratic optimal stochastic control problems with random coefficients:linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim. 42, 5375 (2003), 
[20] 
Tang, S:Dynamic programming for general linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 53, 10821106 (2015), 
[21] 
Tevzadze, R:Solvability of backward stochastic differential equations with quadratic growth. Stochastic. Process. Appl. 118, 503515 (2008), 
show all references
References:
[1] 
Bismut, J.M:Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384404(1973), 
[2] 
Bismut, J.M:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control. Optim. 14, 419444 (1976), 
[3] 
Briand, P.H, Hu, Y:BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Fields. 136, 604618 (2006), 
[4] 
Briand, P.H, Hu, Y:Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields. 141, 543567 (2008), 
[5] 
Chen, S.:Introduction to Modern Partial Differential Equations. Science Press, Beijing (2005), 
[6] 
Du, K.:Backward Stochastic Differential Equations and Their Applications. Dissertation, School of Mathematical Sciences, Fudan University (2011), 
[7] 
Du, K, Chen, S:Backward stochastic partial differential equations with quadratic growth. J. Math. Anal. Appl. 419, 447468 (2012), 
[8] 
Du, K, Qiu, J, Tang, S:L^{p} theory for superparabolic backward stochastic partial differential equations in the whole space. Appl. Math. Optim. 65, 175219 (2012), 
[9] 
Du, K, Tang, S:Strong solution of backward stochastic partial differential equations in C^{2} domains. Probab. Theory Related Fields. 154, 255285 (2010), 
[10] 
Du, K, Zhang, Q:Semilinear degenerate backward stochastic partial differential equations and associated forward backward stochastic differential equations. Stoch. Proc. Appl. 123, 16161637 (2013), 
[11] 
El Karoui, N, Hamadène, S:BSDEs and risksensitive control, zerosum and nonzerosum game problems of stochastic functional differential equations. Stochast. Process. Appl. 107, 145169 (2003), 
[12] 
Frei, C, dos Reis, G:A financial market with interacting investors:does an equilibrium exist? Math. Finan. Econ. 4, 161182 (2011), 
[13] 
Fujita, H:On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations. Proc. Sympos. Pure Math. 18, 105113 (1969), 
[14] 
Hu, Y, Tang, S:Multidimensional backward stochastic differential equations of diagonally quadratic generators. Stoch. Proc. Appl. 126, 10661086 (2016), 
[15] 
Kobylanski, M:Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558602 (2000), 
[16] 
Lepeltier, J.P, San Martin, J:Backward stochastic differential equations with continuous coefficients. Statist. Probab. Lett. 32, 425430 (1997), 
[17] 
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 5561 (1990), 
[18] 
Qiu, J, Tang, S:Maximum principle for quasilinear backward stochastic partial differential equations. J. Funct. Anal. 262, 24362480 (2012), 
[19] 
Tang, S:General linear quadratic optimal stochastic control problems with random coefficients:linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim. 42, 5375 (2003), 
[20] 
Tang, S:Dynamic programming for general linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 53, 10821106 (2015), 
[21] 
Tevzadze, R:Solvability of backward stochastic differential equations with quadratic growth. Stochastic. Process. Appl. 118, 503515 (2008), 
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