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 Published  March 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)

[1]

Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028

[2]

Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285

[3]

John A. D. Appleby, John A. Daniels. Exponential growth in the solution of an affine stochastic differential equation with an average functional and financial market bubbles. Conference Publications, 2011, 2011 (Special) : 91-101. doi: 10.3934/proc.2011.2011.91

[4]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control and Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013

[5]

Alain Bensoussan, Jens Frehse. On diagonal elliptic and parabolic systems with super-quadratic Hamiltonians. Communications on Pure and Applied Analysis, 2009, 8 (1) : 83-94. doi: 10.3934/cpaa.2009.8.83

[6]

Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683

[7]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[8]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002

[9]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075

[10]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803

[11]

Joscha Diehl, Jianfeng Zhang. Backward stochastic differential equations with Young drift. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 5-. doi: 10.1186/s41546-017-0016-5

[12]

Wojciech Kryszewski, Dorota Gabor, Jakub Siemianowski. The Krasnosel'skii formula for parabolic differential inclusions with state constraints. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 295-329. doi: 10.3934/dcdsb.2018021

[13]

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

[14]

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

[15]

Oana Pocovnicu. Explicit formula for the solution of the Szegö equation on the real line and applications. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 607-649. doi: 10.3934/dcds.2011.31.607

[16]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065

[17]

Ruy Coimbra Charão, Juan Torres Espinoza, Ryo Ikehata. A second order fractional differential equation under effects of a super damping. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4433-4454. doi: 10.3934/cpaa.2020202

[18]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040

[19]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565

[20]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905

 Impact Factor: 

Metrics

  • PDF downloads (32)
  • HTML views (390)
  • Cited by (0)

Other articles
by authors

[Back to Top]