Refined Solutions of Time Inhomogeneous Optimal Stopping Games via Dirichlet Form

The properties of value functions of time inhomogeneous optimal stopping problem and zero-sum game (Dynkin game) are studied through time dependent Dirichlet form. Under the absolute continuity condition on the transition function of the underlying diffusion process and some other assumptions, the refined solutions without exceptional starting points are proved to exist, and the value functions of the optimal stopping and zero-sum game, which are finely and cofinely continuous, are characterized as the solutions of some variational inequalities, respectively

On two-sided controls of a linear diffusion

siirretty Doriast

Finite Horizon Time Inhomogeneous Singular Control Problem of One-dimensional Diffusion via Dynkin Game

The Hamilton-Jacobi-Bellman equation (HJB) associated with the time inhomogeneous singular control problem is a parabolic partial differential equation, and the existence of a classical solution is usually difficult to prove. In this paper, a class of finite horizon stochastic singular control problems of one dimensional diffusion is solved via a time inhomogeneous zero-sum game (Dynkin game). The regularity of the value function of the Dynkin game is investigated, and its integrated form coincides with the value function of the singular control problem. We provide conditions under which a classical solution to the associated HJB equation exists, thus the usual viscosity solution approach is avoided. We also show that the optimal control policy is to reflect the diffusion between two time inhomogeneous boundaries. For a more general terminal payoff function, we showed that the optimal control involves a possible impulse at maturity

Controlled diffusion processes

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Minimum Guaranteed Payments and Costly Cancellation Rights: A Stopping Game Perspective

We consider the valuation and optimal exercise policy of a δ- penalty minimum guaranteed payment option in the case where the value of the underlying dividend-paying asset follows a linear diffusion. We characterize both the value and optimal exercise policy of the considered game option explicitly and demonstrate that increased volatility increases the value of the option and postpones exercise by expanding the continuation region where exercising is suboptimal. An interesting and natural implication of this finding is that the value of the embedded cancellation rights of the issuer increase as volatility increases.minimum guaranteed payment, δ-penalty options, Dynkin games, linear diffusions

Some applications of optimal stopping and control in finance and economics

In this thesis, we consider some applications of optimal stopping and control problems in specific scenarios. In Chapter 1, a review of the established general results is provided. In Chapter 2, we study a mathematical model capturing the support/ resistance line method (a technique in technical analysis) where the underlying stock price transitions between two states of nature in a path-dependent manner. For optimal stopping problems with respect to a general class of reward functions and dynamics, using probabilistic methods, we show that the value function is C1 and solves a general free boundary problem. Moreover, for a wide range of utilities, we prove that the best time to buy and sell the stock is obtained by solving free boundary problems corresponding to two linked optimal stopping problems. We use this to numerically compute optimal trading strategies and compare the strategies with the standard trading rule to investigate the viability of this form of technical analysis. In Chapter 3, the model studied in Chapter 2 is extended by adding a partial reflection boundary and an additional regime (the 0 regime). In Chapter 4, we study a two dimensional continuous-time infinite horizon singular control problem related with the optimal management of inventory and production. The primary source of production is modeled as an uncontrolled one-dimensional diffusion process with general dynamics. By controlling the accumulated secondary source of production and output, which are both finite variation processes, we aim to optimise the inventory process under a general concave running reward function and maximise the profit generated from the production. By solving the associated Dynkin game, we obtain two non-intersecting bounded and monotone free-boundaries where one is directly computable and the other is characterised by a free-boundary problem with smooth-pasting conditions. By restricting the volatility term of the diffusion to linear functions with no intercepts, desired smoothness of the value function is obtained by utilising its viscosity property. This leads to the verification of the proposed candidate optimal control that keeps the state process within the inaction set by reflecting the inventory process at the free-boundaries with the minimum effort