Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations
Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations
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DOI：
10.1007/s00245-019-09641-7
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发表时间：
2019-12-20
影响因子：
1.8
通讯作者：
Yong, Jiongmin
中科院分区：
文献类型：
--
作者：
Wang, Hanxiao;Sun, Jingrui;Yong, Jiongmin
对于欧式或有债权的 FT 可测量收益，递归效用过程/动态风险度量可以通过后向随机微分方程 (BSDE) 的适应解来描述。然而，对于 FT 可测量的随机过程（称为位置过程，不一定是 F 适应的），模仿 BSDE 的方法将导致时间不一致的递归效用/动态风险测量。研究发现，更合适的方法是使用向后随机 Volterra 积分方程 (BSVIE) 的自适应解。相应的概念分别称为均衡递归效用和均衡动态风险测度。受此启发，本文关注的是 BSVIE，其生成器允许二次增长（在 Z(t, s) 中）。所谓的适应解和适应M-解的存在性和唯一性被建立。还建立了针对所谓的 I 型 BSVIE 的适应性解决方案的比较定理。作为这些结果的结果，构建了一些一般的连续时间均衡动态风险度量和均衡递归效用过程。
For an FT -measurable payoff of a European type contingent claim, the recursive utility process/dynamic risk measure can be described by the adapted solution to a backward stochastic differential equation (BSDE). However, for an FT -measurable stochastic process (called a position process, not necessarily F-adapted), mimicking BSDE's approach will lead to a time-inconsistent recursive utility/dynamic risk measure. It is found that a more proper approach is to use the adapted solution to a backward stochastic Volterra integral equation (BSVIE). The corresponding notions are called equilibrium recursive utility and equilibrium dynamic risk measure, respectively. Motivated by this, the current paper is concerned with BSVIEs whose generators are allowed to have quadratic growth (in Z(t, s)). The existence and uniqueness for both the so-called adapted solutions and adapted M-solutions are established. A comparison theorem for adapted solutions to the so-called Type-I BSVIEs is established as well. As consequences of these results, some general continuous-time equilibrium dynamic risk measures and equilibrium recursive utility processes are constructed.