BELLMAN EQUATIONS OF RISK-SENRSITIVE STOCHASTIC AND THEIR APPLICATIONS

风险敏感随机贝尔曼方程及其应用

• 批准号: 13440033
• 负责人: NAGAI Hideo
• 金额: $ 7.62万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440033/
• 关键词: Bellman equations; Portfolio optimization; Maximum principle; Log Sobolev inequalities; Exponential hedge; Viscosity solutions; Critical surface models; semi-classical limits; ポートフェリオ最適化; ハミルトン・ヤコビ方程式; 緩和現象; コルモゴロフ方程式; 最適戦略; ウィーナー空間; 非完備市場 /

1. We considered risk-sensitive portfolio optimization problems on infinite time horizon for linear Gaussian models and general factormodels. Proving existence of solutions of ergodic type Bellman equations we got the results constructing explicitly the optimal strategies from the solutions. As for linear Gaussian models we got the same results in the case of partial information as well by only using the informations of security prices.2. In the case of partial information, using the information of only security prices, we obtained maximum principle as necessary conditions for optimality for the problems on a finite time horizon3. In the above case we showed that optimal strategies could be expressed explicitly by using the solution of Bellman equation with degenerate coefficients for conditionally Gaussian models4. We showed semi-classical behavior of the minimum eigenvalues of Schrodinger operators on Wiener space can be captured in a similar way to the case of finite dimensions. By … More using similar idea we proved rough lower estimates holds for the minimum eigenvalues of the operators on path spaces (not pinned) on Riemannian manifolds. We also proved, by considering semi-classical limits on the pinned pathe space on Lie groups, that it implies that harmonic forms vanishes5. We studied estimates of log derivatives of the heat kernels on Riemannian manifolds in which curvatures rapidly decrease enough and proved log Sobolev inequalities on path spaces. We also studied relationships between Brownian rough path and weak type poincare inequalities.6. We studied optimization problems concerning exponential hedging in mathematical finance. In particular we calculated asymptotic expansion of the backward stochastic differential equations with respect to small parameter and obtained asymptotics of the optimal controls7. We constructed optimal portfolio by getting higher order differentiability of the solutions of nonlinear partial differential equations arising from mathematical finance8. We got interested in solving optimization problem by the methods of convex duality in mathematical finance and extended known. results in applying the methods to the case of partial information, or super hedging under constraints with respect to delta9. We got the results on exsistence and uniqueness of viscosity solutions by deriving Euler equations as singular limits of minimum elements of minimization problems of functionals topologically equivalent. We got the Holder estimates of Lp viscosity solutions of fully nonlinear elliptic partial differential equation with super-linear growth with respect to first order derivatives.10. We discussed hydrodynamic limits of critical surface models on walls and derived variational inequalities of evolution type. We also derived Alt-Caffarelli variational problems by proving large deviation principles for equilibrium systems of the critical surfaces with pinning. Less

1.考虑了无限时间范围上线性高斯模型和一般因子模型下的风险敏感投资组合优化问题。证明了遍历型Bellman方程解的存在性，得到了由解构造最优策略的显式结果。对于线性高斯模型，仅利用证券价格信息，在部分信息的情况下也得到了相同的结果.在部分信息的情况下，仅利用证券价格的信息，得到了有限时间水平上最优性的必要条件-极大值原理。在上面的例子中，我们证明了最优策略可以通过使用条件高斯模型的具有退化系数的Bellman方程的解来显式地表达4。我们证明了维纳空间上薛定谔算子最小本征值的半经典行为可以以与有限维情况类似的方式捕获。通过 ...更多信息 利用类似的思想，我们证明了黎曼流形上路径空间（非钉住）上算子的最小特征值的粗糙下估计成立。我们还证明，通过考虑李群上的pinned pathe空间的半经典极限，它意味着调和形式消失。研究了曲率迅速减小到足够小的黎曼流形上热核的对数导数的估计，证明了路空间上的对数Sobolev不等式。研究了布朗粗糙路与弱型Poincare不等式之间的关系.研究了数理金融中指数套期保值的最优化问题。特别地，我们计算了倒向随机微分方程关于小参数的渐近展开，并得到了最优控制的渐近性。我们利用数学金融学中非线性偏微分方程解的高阶可微性构造最优投资组合8。本文利用金融学中的凸对偶方法和推广的已知方法来解决最优化问题。导致将所述方法应用于部分信息的情况，或者在相对于Δ θ的约束下的超对冲。通过将Euler方程作为拓扑等价泛函极小化问题最小元的奇异极限，得到了粘性解的存在唯一性结果.我们得到了具有超线性增长的完全非线性椭圆型偏微分方程Lp粘性解关于一阶导数的保持器估计.讨论了壁面临界面模型的水动力极限，导出了发展型变分不等式。我们还通过证明具有钉扎的临界面平衡系统的大偏差原理导出了Alt-Caffarelli变分问题。少

项目成果

Hideo NAGAI: "Optimal strategies for risk-sensitive portfolio optimization problems for general factor models"SIAM Journal on Control and Optimization. 41. 1779-1800 (2003)

Hideo NAGAI：“一般因子模型的风险敏感投资组合优化问题的最优策略”SIAM 控制与优化杂志。

Shigeki Aida: "Semiclassical limit of the lowest eigenvalue of a Schrodinger operator on a Wiener space"Journal of Functional Analysis. 203. 401-424 (2003)

Shigeki Aida：“维纳空间上薛定谔算子最低特征值的半经典极限”泛函分析杂志。

K.Kuroda: "Ergodic type Bellman equation of risk sensitive control and portfolio optimization on infinite time horizon"Optimal Control and Partial Differential Equations, Eds.Menaldi et al. IOS press, Amsterdam. 530-538 (2001)

K.Kuroda：“无限时间范围内风险敏感控制和投资组合优化的遍历型贝尔曼方程”最优控制和偏微分方程，Eds.Menaldi 等人。

K.Kuroda: "Risk-sensitive portfolio optimization on infinite time horizon"Stochastics and Stochastics Reports. 73. 309-332 (2002)

K.Kuroda：“无限时间范围内的风险敏感投资组合优化”随机指标和随机指标报告。

H.Ishii: "Asymptotic analysis for a class of infinite systems of first-order PDE : nonlinear parabolic PDE in the singular limit"Communication on Partial differential Equations. 28. 409-438 (2003)

H.Ishii：“一类无限一阶 PDE 系统的渐近分析：奇异极限下的非线性抛物线 PDE”偏微分方程通讯。