Regularity of the Value Function for a Two-Dimensional Singular Stochastic Control Problem

1989 ◽  
Vol 27 (4) ◽  
pp. 876-907 ◽  
Author(s):  
H. Mete Soner ◽  
Shreve E. Shreve
Keyword(s):  
Control Problem ◽  
Stochastic Control ◽  
Value Function ◽  
Two Dimensional ◽  
Singular Stochastic Control ◽  
Stochastic Control Problem ◽  
The Value Function

scholarly journals A multidimensional singular stochastic control problem on a finite time horizon

2015 ◽  
Vol 69 (1) ◽  
Author(s):  
Marcin Boryc ◽  
Łukasz Kruk
Keyword(s):  
Control Problem ◽  
Stochastic Control ◽  
Finite Time ◽  
Time Horizon ◽  
Generalized Solution ◽  
Singular Stochastic Control ◽  
Stochastic Control Problem ◽  
Second Derivatives ◽  
Finite Time Horizon ◽  
The Value Function

AbstractA singular stochastic control problem in n dimensions with timedependent coefficients on a finite time horizon is considered. We show that the value function for this problem is a generalized solution of the corresponding HJB equation with locally bounded second derivatives with respect to the space variables and the first derivative with respect to time. Moreover, we prove that an optimal control exists and is unique

scholarly journals Optimal dividend and reinsurance in the presence of two reinsurers

2016 ◽  
Vol 53 (2) ◽  
pp. 554-571 ◽  
Author(s):  
Mi Chen ◽  
Kam Chuen Yuen
Keyword(s):  
Control Problem ◽  
Transaction Costs ◽  
Stochastic Control ◽  
Value Function ◽  
Optimal Reinsurance ◽  
Stochastic Control Problem ◽  
Optimal Dividend ◽  
The Impact ◽  
The Value Function ◽  
Limit Diffusion

Abstract In this paper the optimal dividend (subject to transaction costs) and reinsurance (with two reinsurers) problem is studied in the limit diffusion setting. It is assumed that transaction costs and taxes are required when dividends occur, and that the premiums charged by two reinsurers are calculated according to the exponential premium principle with different parameters, which makes the stochastic control problem nonlinear. The objective of the insurer is to determine the optimal reinsurance and dividend policy so as to maximize the expected discounted dividends until ruin. The problem is formulated as a mixed classical-impulse stochastic control problem. Explicit expressions for the value function and the corresponding optimal strategy are obtained. Finally, a numerical example is presented to illustrate the impact of the parameters associated with the two reinsurers' premium principle on the optimal reinsurance strategy.

scholarly journals A multidimensional singular stochastic control problem on a finite time horizon

2015 ◽  
Vol 69 (1) ◽  
pp. 23
Author(s):  
Marcin Boryc ◽  
Łukasz Kruk
Keyword(s):  
Control Problem ◽  
Stochastic Control ◽  
Finite Time ◽  
Time Horizon ◽  
Generalized Solution ◽  
Singular Stochastic Control ◽  
Stochastic Control Problem ◽  
Second Derivatives ◽  
Finite Time Horizon ◽  
The Value Function

A singular stochastic control problem in n dimensions with timedependent coefficients on a finite time horizon is considered. We show that the value function for this problem is a generalized solution of the corresponding HJB equation with locally bounded second derivatives with respect to the space variables and the first derivative with respect to time. Moreover, we prove that an optimal control exists and is unique.

scholarly journals Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

2019 ◽  
Vol 24 (1) ◽  
pp. 71-123 ◽  
Author(s):  
Tiziano De Angelis
Keyword(s):  
Stochastic Control ◽  
Value Function ◽  
Partial Information ◽  
Point Of View ◽  
Two Dimensional ◽  
Singular Stochastic Control ◽  
State Dependent ◽  
Optimal Dividend ◽  
Analytical Point ◽  
The Value Function

Abstract We study the optimal dividend problem for a firm’s manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a two-dimensional degenerate diffusion whose first component is singularly controlled. Moreover, the process is absorbed when its first component hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with ‘creation’. One key feature of the stopping problem is that creation occurs at a state-dependent rate of the ‘local time’ of an auxiliary two-dimensional reflecting diffusion.

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