英文标题：
《Stochastic control for a class of nonlinear kernels and applications》
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作者：
Dylan Possama\\\"i and Xiaolu Tan and Chao Zhou
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最新提交年份：
2017
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英文摘要：
We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimisation, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalisations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semi-martingale characterisation of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in [86]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of [71], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).
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中文摘要：
我们考虑一类非线性核的随机控制问题。更准确地说，我们感兴趣的问题在于优化倒向随机微分方程（BSDE）的解，通过一组可能非支配的概率测度。由于BSDE是传统（线性）期望的非线性推广，该问题可以理解为一系列非线性期望的随机控制，或等效的非线性核。我们的第一个主要贡献是在一个抽象的环境中证明这个控制问题的动态规划原理，然后我们用它来提供值函数的半鞅特征。接下来，我们将探讨我们的结果的几个应用。我们首先得到了二阶BSDE（如[86]中介绍的）的适定性结果，它不需要对终端条件和生成器进行任何正则性假设。然后，我们证明了稳健环境下的非线性可选分解，扩展了[71]的最新结果，然后我们利用这些结果在不确定、不完全和非线性金融市场中获得了超套期对偶。最后，在额外的正则性假设下，我们将值函数与适当的路径相关偏微分方程（PPDE）的粘性解联系起来。
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分类信息：

一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Mathematics 数学
二级分类：Optimization and Control 优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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