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This paper analyzes a class of infinite-time-horizon stochastic games with singular controls motivated from the partially reversible problem. It provides an explicit solution for the mean-field game (MFG) and presents sensitivity analysis to compare the solution for the MFG with that for the single-agent control problem. It shows that in the MFG, model parameters not only affect the optimal strategies as in the single-agent case, but also influence the equilibrium price. It then establishes that the solution to the MFG is an $epsilon$-Nash Equilibrium to the corresponding $N$-player game, with $epsilon=Oleft(frac{1}{sqrt N}right)$.
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This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time-inconsistent stopping control problems under general multi-dimensional controlled diffusion model and propose a formal definition of their equilibriums. We show that an admissible pair $(hat{u},C)$ of control-stopping policy is equilibrium if and only if the axillary function associated to it solves the extended HJB system. We provide almost equivalent conditions to the boundary term of this extended HJB system, which is related to the celebrated smooth fitting principles. As applications of our theoretical results, we develop an investment-withdrawal decision model for time-inconsistent decision makers in infinite time horizon. We provide two concrete examples, one of which includes constant proportion investment with one side threshold withdrawal strategy as equilibrium; in another example, all strategies with constant proportion investment are proved to be irrational, no matter what the withdrawal strategy is.
We consider an optimal stochastic impulse control problem over an infinite time horizon motivated by a model of irreversible investment choices with fixed adjustment costs. By employing techniques of viscosity solutions and relying on semiconvexity arguments, we prove that the value function is a classical solution to the associated quasi-variational inequality. This enables us to characterize the structure of the continuation and action regions and construct an optimal control. Finally, we focus on the linear case, discussing, by a numerical analysis, the sensitivity of the solution with respect to the relevant parameters of the problem.
In this paper, we study an irreversible investment problem under Knightian uncertainty. In a general framework, in which Knightian uncertainty is modeled through a set of multiple priors, we prove existence and uniqueness of the optimal investment plan, and derive necessary and sufficient conditions for optimality. This allows us to construct the optimal policy in terms of the solution to a stochastic backward equation under the worst-case scenario. In a time-homogeneous setting - where risk is driven by a geometric Brownian motion and Knightian uncertainty is realized through a so-called k-ignorance - we are able to provide the explicit form of the optimal irreversible investment plan.
This paper considers a life-time consumption-investment problem under the Black-Scholes framework, where the investors consumption rate is subject to a lower bound constraint that linearly depends on the investors wealth. Due to the state-dependent control constraint, the standard stochastic control theory cannot be directly applied to our problem. We overcome this obstacle by examining an equivalent problem that does not impose state-dependent control constraint. It is shown that the value function is a third-order continuously differentiable function by using differential equation approaches. The feedback form optimal consumption and investment strategies are given. According to our findings, if the investor is more concerned with long-term consumption than short-term consumption, then she should, regardless of her financial condition, always consume as few as possible; otherwise, her optimal consumption strategy is state-dependent: consuming optimally when her financial condition is good, and consuming at the lowest possible rate when her financial situation is bad.
A new definition of continuous-time equilibrium controls is introduced. As opposed to the standard definition, which involves a derivative-type operation, the new definition parallels how a discrete-time equilibrium is defined, and allows for unambiguous economic interpretation. The terms strong equilibria and weak equilibria are coined for controls under the new and the standard definitions, respectively. When the state process is a time-homogeneous continuous-time Markov chain, a careful asymptotic analysis gives complete characterizations of weak and strong equilibria. Thanks to Kakutani-Fans fixed-point theorem, general existence of weak and strong equilibria is also established, under additional compactness assumption. Our theoretic results are applied to a two-state model under non-exponential discounting. In particular, we demonstrate explicitly that there can be incentive to deviate from a weak equilibrium, which justifies the need for strong equilibria. Our analysis also provides new results for the existence and characterization of discrete-time equilibria under infinite horizon.
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