效用最大化中的动态鲁棒对偶

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摘要翻译：
凸对偶理论的一个著名的金融应用给出了以下两个量之间的显式关系：（一）在风险资产价格过程为半线性模型的市场中，最优终端财富$x^*(T):=X_{varphi^*}(T)$使可容许投资组合$varphi(T)，0\leqt\leqt$产生的终端财富$X_{varphi^*}(T)$的期望效用最大化问题的X_{varphi^*}(T)$；(ii)对偶问题的最优方案$\frac{dQ^*}{dP}$在一个等价局部鞅测度族$q$上使$\frac{dQ}{dP}$的期望$v$-值最小，其中$v$是凹函数$u$的凸共轭函数。本文考虑了用IT-O-L-Evy过程建模的市场。第一部分利用随机控制理论中的极大值原理，将上述关系推广到a\emph{动态}关系，对[0,t]$中的所有$t\都有效。我们特别证明了原始问题的最优伴随过程与最优密度过程重合，对偶问题的最优伴随过程与最优财富过程$0\leqt\leqt$重合。在终端时间为$t=t$的情况下，我们恢复了上面的经典对偶联系。此外，我们还得到了最优投资组合$\varphi^*$与最优测度$q^*$之间的显式关系。我们还得到了一个最优方案的存在性等价于一个相关的$T$-索赔的可复制性。在第二部分中，我们给出了(i)和(ii)中优化问题的鲁棒（模型不确定性）版本，并证明了它们之间类似的动态关系。特别是，我们展示了如何从其中一个问题的解到另一个问题的解。我们用明确的例子说明了结果。
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英文标题：
《Dynamic robust duality in utility maximization》
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作者：
Bernt {\O}ksendal and Agn\`es Sulem
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最新提交年份：
2015
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Portfolio Management        项目组合管理
分类描述：Security selection and optimization, capital allocation, investment strategies and performance measurement
证券选择与优化、资本配置、投资策略与绩效评价
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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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英文摘要：
  A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities:   (i) The optimal terminal wealth $X^*(T) : = X_{\varphi^*}(T)$ of the problem to maximize the expected $U$-utility of the terminal wealth $X_{\varphi}(T)$ generated by admissible portfolios $\varphi(t), 0 \leq t \leq T$ in a market with the risky asset price process modeled as a semimartingale;   (ii) The optimal scenario $\frac{dQ^*}{dP}$ of the dual problem to minimize the expected $V$-value of $\frac{dQ}{dP}$ over a family of equivalent local martingale measures $Q$, where $V$ is the convex conjugate function of the concave function $U$.   In this paper we consider markets modeled by It\^o-L\'evy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a \emph{dynamic} relation, valid for all $t \in [0,T]$. We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process, $0 \leq t \leq T$. In the terminal time case $t=T$ we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio $\varphi^*$ and the optimal measure $Q^*$. We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related $T$-claim.   In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples.
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PDF链接：
https://arxiv.org/pdf/1304.5040

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关键词：效用最大化 最大化 frac 财富 对偶 问题 problem