We consider a singular stochastic control problem, which is called the Monotone Follower Stochastic Control Problem and give sufficient conditions for the existence and uniqueness of a local-time type optimal control. To establish this result we use a methodology that has not been employed to solve singular control problems. We first confine ourselves to local time strategies. Then we apply a transformation to the total reward accrued by reflecting the diffusion at a given boundary and show that it is linear in its continuation region. Now, the problem of finding the optimal boundary becomes a non-linear optimization problem: The slope of the linear function and an obstacle function need to be simultaneously maximized. The necessary conditions of optimality come from first order derivative conditions. We show that under some weak assumptions these conditions become sufficient. We also show that the local time strategies are optimal in the class of all monotone increasing controls.
As a byproduct of our analysis, we give sufficient conditions for the value function to be $\mathbf{C}^2$ on all its domain.
We solve two dividend payment problems to show that our sufficient conditions are satisfied by the examples considered in the mainstream literature. We show that our assumptions are satisfied not only when capital of a company is modeled by a Brownian motion with drift but also when we change the modeling assumptions and use a square root process to model the capital.
Existence of optimal controls for SPDE with locally monotone coefficients
研究了由局部单调系数的随机偏微分方程（SPDE）描述的系统在不同外力控制下的最优控制的存在性，这些外力控制是反馈控制.为了达到我们的目的，我们采用了H. Lisei，（Existence of optimal and non-最优控制for the stochastic Navier -斯托克斯方程，Nonlinear Analysis，51，（2002）95-118），其中研究了随机Navier-斯托克斯方程的最优控制的存在性。本文所得结果可应用于证明各种类型受控SPDE最优控制的存在性，如：随机非局部方程和局部单调的随机半线性方程;我们也将结果应用于单调方程，如随机反应扩散方程和随机线性方程.
On the dynamic consistency of hierarchical risk-averse decision problems
In this paper, we consider a risk-averse decision problem for controlled-diffusion processes, with dynamic risk measures, in which there are two risk-averse decision makers (i.e., { \it leader} and { \it follower}) with different risk-averse related responsibilities and information. Moreover, we assume that there are two objectives that these decision makers are expected to achieve. That is, the first objective being of { \it stochastic controllability} type that describes an acceptable risk-exposure set vis-á-vis some uncertain future payoff, and while the { \it second one} is making sure the…
Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operators
Sapienza Università di Roma（罗马大学）；Technische Universität Berlin（柏林工业大学）
