标题：Parabolic equations with quadratic growth in ℝ n
作者：Bensoussan A.; Frehse J.; Peng S.; Yam S.C.P.
作者机构：[Bensoussan, A] International Center for Decision and Risk Analysis, Jindal School of Management, University of Texas at Dallas, Richardson, United St 更多
通讯作者地址：[Bensoussan, A] International Center for Decision and Risk Analysis, Jindal School of Management, University of Texas at DallasUnited States;
来源：Computational Methods in Applied Sciences
摘要：We study here quasi-linear parabolic equations with quadratic growth in ℝ n . These parabolic equations are at the core of the theory of PDE; see Friedman (Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964) , Ladyzhenskaya et al. (Translations of Mathematical Monographs. AMS, 1968)  for details. However, for the applications to physics and mechanics, one deals mostly with boundary value problems. The boundary is often taken to be bounded and the solution is bounded. This brings an important simplification. On the other hand, stochastic control theory leads mostly to problems in ℝ n . Moreover, the functions are unbounded and the Hamiltonian may have quadratic growth. There may be conflicts which prevent solutions to exist. In stochastic control theory, a very important development deals with BSDE (Backward Stochastic Differential Equations). There is a huge interaction with parabolic PDE in ℝ n . This is why, although we do not deal with BSDE in this paper, we use many ideas from Briand and Hu (Probab Theory Relat Fields 141(3–4):543–567, 2008) , Da Lio and Ley (SIAM J Control Optim 45(1):74–106, 2006) , Karoui et al. (Backward stochastic differential equations and applications, Princeton BSDE Lecture Notes, 2009) , Kobylanski (Ann Probab 28(2):558–602, 2000) . Our presentation provided here is slightly innovative. © Springer International Publishing AG, part of Springer Nature 2019.