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The Cauchy problem of Backward Stochastic Super-Parabolic 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, 384-404(1973), |
[2] |
Bismut, J.-M:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control. Optim. 14, 419-444 (1976), |
[3] |
Briand, P.H, Hu, Y:BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Fields. 136, 604-618 (2006), |
[4] |
Briand, P.H, Hu, Y:Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields. 141, 543-567 (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, 447-468 (2012), |
[8] |
Du, K, Qiu, J, Tang, S:Lp theory for super-parabolic backward stochastic partial differential equations in the whole space. Appl. Math. Optim. 65, 175-219 (2012), |
[9] |
Du, K, Tang, S:Strong solution of backward stochastic partial differential equations in C2 domains. Probab. Theory Related Fields. 154, 255-285 (2010), |
[10] |
Du, K, Zhang, Q:Semi-linear degenerate backward stochastic partial differential equations and associated forward backward stochastic differential equations. Stoch. Proc. Appl. 123, 1616-1637 (2013), |
[11] |
El Karoui, N, Hamadène, S:BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stochast. Process. Appl. 107, 145-169 (2003), |
[12] |
Frei, C, dos Reis, G:A financial market with interacting investors:does an equilibrium exist? Math. Finan. Econ. 4, 161-182 (2011), |
[13] |
Fujita, H:On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations. Proc. Sympos. Pure Math. 18, 105-113 (1969), |
[14] |
Hu, Y, Tang, S:Multi-dimensional backward stochastic differential equations of diagonally quadratic generators. Stoch. Proc. Appl. 126, 1066-1086 (2016), |
[15] |
Kobylanski, M:Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558-602 (2000), |
[16] |
Lepeltier, J.P, San Martin, J:Backward stochastic differential equations with continuous coefficients. Statist. Probab. Lett. 32, 425-430 (1997), |
[17] |
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55-61 (1990), |
[18] |
Qiu, J, Tang, S:Maximum principle for quasi-linear backward stochastic partial differential equations. J. Funct. Anal. 262, 2436-2480 (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, 53-75 (2003), |
[20] |
Tang, S:Dynamic programming for general linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 53, 1082-1106 (2015), |
[21] |
Tevzadze, R:Solvability of backward stochastic differential equations with quadratic growth. Stochastic. Process. Appl. 118, 503-515 (2008), |
show all references
References:
[1] |
Bismut, J.M:Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384-404(1973), |
[2] |
Bismut, J.-M:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control. Optim. 14, 419-444 (1976), |
[3] |
Briand, P.H, Hu, Y:BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Fields. 136, 604-618 (2006), |
[4] |
Briand, P.H, Hu, Y:Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields. 141, 543-567 (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, 447-468 (2012), |
[8] |
Du, K, Qiu, J, Tang, S:Lp theory for super-parabolic backward stochastic partial differential equations in the whole space. Appl. Math. Optim. 65, 175-219 (2012), |
[9] |
Du, K, Tang, S:Strong solution of backward stochastic partial differential equations in C2 domains. Probab. Theory Related Fields. 154, 255-285 (2010), |
[10] |
Du, K, Zhang, Q:Semi-linear degenerate backward stochastic partial differential equations and associated forward backward stochastic differential equations. Stoch. Proc. Appl. 123, 1616-1637 (2013), |
[11] |
El Karoui, N, Hamadène, S:BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stochast. Process. Appl. 107, 145-169 (2003), |
[12] |
Frei, C, dos Reis, G:A financial market with interacting investors:does an equilibrium exist? Math. Finan. Econ. 4, 161-182 (2011), |
[13] |
Fujita, H:On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations. Proc. Sympos. Pure Math. 18, 105-113 (1969), |
[14] |
Hu, Y, Tang, S:Multi-dimensional backward stochastic differential equations of diagonally quadratic generators. Stoch. Proc. Appl. 126, 1066-1086 (2016), |
[15] |
Kobylanski, M:Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558-602 (2000), |
[16] |
Lepeltier, J.P, San Martin, J:Backward stochastic differential equations with continuous coefficients. Statist. Probab. Lett. 32, 425-430 (1997), |
[17] |
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55-61 (1990), |
[18] |
Qiu, J, Tang, S:Maximum principle for quasi-linear backward stochastic partial differential equations. J. Funct. Anal. 262, 2436-2480 (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, 53-75 (2003), |
[20] |
Tang, S:Dynamic programming for general linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 53, 1082-1106 (2015), |
[21] |
Tevzadze, R:Solvability of backward stochastic differential equations with quadratic growth. Stochastic. Process. Appl. 118, 503-515 (2008), |
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