Large Time Behavior of Solutions to SemiLinear Equations with Quadratic Growth in the Gradient
Large Time Behavior of Solutions to SemiLinear Equations with Quadratic Growth in the Gradient
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DOI：
10.1137/13094311x
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发表时间：
2013-07
期刊：
影响因子：
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通讯作者：
Scott Robertson;Hao Xing
中科院分区：
文献类型：
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作者：
Scott Robertson;Hao Xing
本文研究了具有梯度二次非线性的半线性柯西问题解的大时间行为。所考虑的柯西问题具有一般状态空间，并且可能在状态空间的边界上退化。获得两种类型的大时间行为：(i) 解及其梯度的逐点收敛，以及 (ii) 相关后向随机微分方程解的收敛。当状态空间为$\mathbb{R}^d$或正定矩阵空间时，两种类型的收敛都是在系数增长条件下获得的。这些大时间收敛结果可直接应用于风险敏感控制和长期投资组合选择问题。
This paper studies the large time behavior of solutions to semilinear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered has a general state space and may degenerate on the boundary of the state space. Two types of large time behavior are obtained: (i) pointwise convergence of the solution and its gradient and (ii) convergence of solutions to associated backward stochastic differential equations. When the state space is $\mathbb{R}^d$ or the space of positive definite matrices, both types of convergence are obtained under growth conditions on coefficients. These large time convergence results have direct applications in risk-sensitive control and long-term portfolio choice problems.