# American Institute of Mathematical Sciences

January  2019, 4: 3 doi: 10.1186/s41546-019-0037-3

## 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

Received  November 26, 2018 Revised  March 12, 2019

Fund Project: Both Qiu and Tang acknowledge research supported by the National Science Foundation of China (Grants Nos. 11631004 and 11171076), by the Science and Technology Commission, Shanghai Municipality (Grant No. 14XD1400400), and by the Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University.

The paper is devoted to the Cauchy problem of backward stochastic superparabolic equations with quadratic growth. We prove two Itô formulas in the whole space. Furthermore, we prove the existence of weak solutions for the case of one-dimensional state space, and the uniqueness of weak solutions without constraint on the state space.
Citation: Renzhi Qiu, Shanjian Tang. The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 3-. doi: 10.1186/s41546-019-0037-3
##### 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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