BELLMAN EQUATIONS OF RISK-SENRSITIVE STOCHASTIC AND THEIR APPLICATIONS

1.研究了线性高斯模型和一般因子模型在无限时间域上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，得到了由解显式构造最优策略的结果。对于线性高斯模型，在只使用证券价格信息的情况下，我们也得到了相同的结果。在部分信息的情况下，仅利用证券价格信息，得到了极大值原理作为问题在有限时间水平上最优性的必要条件。在上述情况下，我们证明了最优策略可以通过条件高斯模型4的退化系数的Bellman方程的解来显式表示。我们证明了Wiener空间上薛定谔算子最小本征值的半经典行为可以用类似于有限维情形的方法来捕捉。作者：…此外，利用类似的思想，我们证明了黎曼流形上的路径空间(非钉扎)上算子的最小特征值的粗略下估计是成立的。我们还证明了，通过考虑李群上的Pinted Pathe空间上的半经典极限，这意味着调和形式为零。我们研究了曲率下降足够快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Soblev不等式。我们还研究了布朗粗糙路与弱型Poincare不等式之间的关系。研究了数学金融中指数套期保值的优化问题。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式，并得到了最优控制的渐近性。我们通过获得数学金融中产生的非线性偏微分方程解的高阶可微性来构造最优投资组合。利用数学金融学中的凸对偶方法和推广的已知方法，我们对求解最优化问题产生了兴趣。将这些方法应用于部分信息的情况，或关于Delta9的约束下的超级对冲。将欧拉方程作为拓扑等价泛函极小化问题极小元的奇异极限，得到了粘性解的存在唯一性结果。得到了超线性增长的完全非线性椭圆型偏微分方程解关于一阶导数的Lp粘性解的Holder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。通过证明具有钉扎的临界面平衡系统的大偏差原理，我们还导出了Alt-Caffarelli变分问题。较少

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