Singular Stochastic Control Problem Research Articles

Singular control of (reflected) Brownian motion: a computational method suitable for queueing applications

AbstractMotivated by applications in queueing theory, we consider a class of singular stochastic control problems whose state space is the d-dimensional positive orthant. The original problem is approximated by a drift control problem, to which we apply a recently developed computational method that is feasible for dimensions up to $$d=30$$ d = 30 or more. To show that nearly optimal solutions are obtainable using this method, we present computational results for a variety of examples, including queueing network examples that have appeared previously in the literature.

A singular stochastic control problem with direction switching cost

We introduce a new class of singular stochastic control problems in which the process controller not only chooses the push intensity, at a price proportional to the displacement caused by his action, but he can also change the allowable control direction, paying a fixed cost for each such switching. Singular control of the one-dimensional Brownian motion with quadratic instantaneous cost function and costly direction switching on the infinite time horizon is analyzed in detail, leading to a closed-form solution. This example is used as an illustration of qualitative differences between the class of problems considered here and classic singular stochastic control.

Optimal execution with multiplicative price impact and incomplete information on the return

We study an optimal liquidation problem with multiplicative price impact in which the trend of the asset price is an unobservable Bernoulli random variable. The investor aims at selling over an infinite time horizon a fixed amount of assets in order to maximise a net expected profit functional, and lump-sum as well as singularly continuous actions are allowed. Our mathematical modelling leads to a singular stochastic control problem featuring a finite-fuel constraint and partial observation. We provide a complete analysis of an equivalent three-dimensional degenerate problem under full information, whose state process is composed of the asset price dynamics, the amount of available assets in the portfolio, and the investor’s belief about the true value of the asset’s trend. Its value function and optimal execution rule are expressed in terms of the solution to a truly two-dimensional optimal stopping problem, whose associated belief-dependent free boundary b triggers the investor’s optimal selling rule. The curve b is uniquely determined through a nonlinear integral equation, for which we derive a numerical solution through an application of the Monte Carlo method. This allows us to understand the value of information in our model as well as the sensitivity of the problem’s solution with respect to the relevant model parameters.

From the Optimal Singular Stochastic Control to the Optimal Stopping for Regime-Switching Processes

This work generalizes the connection between the optimal singular stochastic control problem and the optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the càdlàg space. Such a problem was left open in Haussmann and Suo [SIAM J. Control Optim., 33 (1995), pp. 916–936] and [SIAM J. Control Optim., 33 (1995), pp. 937–959]). In addition, our compactification method can remove the convexity condition on the coefficients used in Dufour and Miller [SIAM J. Control Optim., 43 (2004), pp. 708–730].

Multidimensional singular control and related Skorokhod problem: Sufficient conditions for the characterization of optimal controls

We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The optimization problems concern the minimization of a discounted cost over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. The setting is multidimensional, the drift of the state equation and the costs are convex, the volatility matrix can be constant or linear in the state. Our result applies to a relevant class of linear-quadratic models and it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature.

Learning about profitability and dynamic cash management

We study a dynamic model of a firm whose shareholders learn about its profitability and, face costs of external financing and costs of holding cash. The shareholders' problem involves a notoriously challenging singular stochastic control problem with a two-dimensional degenerate diffusion process. We solve it by means of an explicit construction of its value function, and derive a corporate life-cycle with two stages: a “probation stage” where it is never optimal for the firm to issue new shares, and a “mature stage” where the firm resorts to the market whenever needed. The cash target level is nonmonotonic in the belief about profitability and reaches its highest value on the edge between the two stages. This solution leads to new insights into the firm's volatility and its payout ratio which depend on the firm's stage in its life cycle.

An irreversible investment problem with demand on a finite horizon: The optimal investment boundary analysis

This paper studies a continuous-time, finite horizon, irreversible investment problem where a social planner aims to minimize total expected costs of production capacity and demand. Our model allows for general diffusion dynamics on the demand as well as production capacity process controlled by a nondecreasing process representing the cumulative investment. Mathematically, it is a singular stochastic control problem whose value function satisfies a two-dimensional parabolic variational inequality with gradient constraint. The problem gives rise to a free boundary which stands for the optimal investment boundary. We use partial differential equation (PDE) approach to characterize some features of the free boundary and prove the C2,1 regularity of the value function. To the best of our knowledge, the method to study the monotonicity of free boundary about the time direction is an innovation.

Optimal Stopping, Randomized Stopping, and Singular Control with General Information Flow

The purpose of this paper is twofold. First, we extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available information flow is a filtration with no a priori assumed relation to the filtration of the process. We call these problems optimal stopping and randomized stopping with general information. Second, following an idea of N. V. Krylov [Controlled Diffusion Processes, Springer-Verlag, 2009], we introduce a special singular stochastic control problem with general information and show that this is also equivalent to the partial information optimal stopping and randomized stopping problems. Then we show that the solution of this singular control problem can be expressed in terms of partial information variational inequalities.

Optimal harvesting under marine reserves and uncertain environment

Persistence in the literature is the perception of an inherent tradeoff between ecological conservation and economic harvesting goals. Overexploitation may lead to resource destruction, including extinction. Conservation measures should be decided and implemented. A standard ecological response is to impose no-take areas or marine reserves. In this paper, our objective is to study a harvesting management problem under the constraints of a no-take area, which we formulate mathematically as a bi-dimensional singular stochastic control problem. Using dynamic programming theory, we characterise our value function as the unique solution to a HJB equation. We also describe the optimal harvesting strategy by identifying the harvesting and non-harvesting regions. We show that setting up reserve area not only leads to a better ecological conservation but may also increase the economic benefit for fishing industry in the long run. We further enrich our studies with some numerical analysis, enabling us to get some insightful understanding on the size of the reserve area where no-take policy should be implemented.

Interbank lending with benchmark rates: Pareto optima for a class of singular control games

AbstractWe analyze a class of stochastic differential games of singular control, motivated by the study of a dynamic model of interbank lending with benchmark rates. We describe Pareto optima for this game and show how they may be achieved through the intervention of a regulator, whose policy is a solution to a singular stochastic control problem. Pareto optima are characterized in terms of the solutions to a new class of Skorokhod problems with piecewise‐continuous free boundary. Pareto optimal policies are shown to correspond to the enforcement of endogenous bounds on interbank lending rates. Analytical comparison between Pareto optima and Nash equilibria provides insight into the impact of regulatory intervention on the stability of interbank rates.

Singular Control of the Drift of a Brownian System

We consider a standard Brownian motion whose drift can be increased or decreased in a possibly singular manner. The objective is to minimize an expected functional involving the time-integral of a running cost and the proportional costs of adjusting the drift. The resulting two-dimensional degenerate singular stochastic control problem has interconnected dynamics and it is solved by combining techniques of viscosity theory and free boundary problems. We provide a detailed description of the problem’s value function and of the geometry of the state space, which is split into three regions by two monotone curves. Our main result shows that those curves are continuously differentiable with locally Lipschitz derivative and solve a system of nonlinear ordinary differential equations.

Penalty method for portfolio selection with capital gains tax

AbstractMany finance problems can be formulated as a singular stochastic control problem, where the associated Hamilton‐Jacobi‐Bellman (HJB) equation takes the form of variational inequality, and its penalty approximation equation is linked to a regular control problem. The penalty method, combined with a finite difference scheme, has been widely used to numerically solve singular control problems, and its convergence analysis in literature relies on the uniqueness of solution to the original HJB equation problem. We consider a singular stochastic control problem arising from continuous‐time portfolio selection with capital gains tax, where the associated HJB equation problem admits infinitely many solutions. We show that the penalty method still works and converges to the value function, which is the minimal (viscosity) solution of the HJB equation problem. The key step is to prove that any admissible singular control can be approximated by a sequence of regular controls related to the corresponding penalized equation problem that admits a unique solution. Numerical results are presented to demonstrate the efficiency of the penalty method and to better understand optimal investment strategy in the presence of capital gains tax. Our approach sheds light on the robustness of the penalty method for general singular stochastic control problems.

On a Class of Infinite-Dimensional Singular Stochastic Control Problems

We study a class of infinite-dimensional singular stochastic control problems that might find applications in economic theory and finance. The control process linearly affects an abstract evolution...

Optimal Installation of Solar Panels with Price Impact: A Solvable Singular Stochastic Control Problem

We consider a price-maker company which generates electricity and sells it in the spot market. The company can increase its level of installed power by irreversible installations of solar panels. T...

An Integral Equation Approach to the Irreversible Investment Problem with a Finite Horizon

This paper studies an irreversible investment problem under a finite horizon. The firm expands its production capacity in irreversible investments by purchasing capital to increase productivity. This problem is a singular stochastic control problem and its associated Hamilton–Jacobi–Bellman equation is derived. By using a Mellin transform, we obtain the integral equation satisfied by the free boundary of this investment problem. Furthermore, we solve the integral equation numerically using the recursive integration method and present the graph for the free boundary.

Modelling information flows by Meyer-$\sigma $-fields in the singular stochastic control problem of irreversible investment

In stochastic control problems delicate issues arise when the controlled system can jump due to both exogenous shocks and endogenous controls. Here one has to specify what the controller knows when about the exogenous shocks and how and when she can act on this information. We propose to use Meyer-$\sigma $-fields as a flexible tool to model information flow in such situations. The possibilities of this approach are illustrated first in a very simple linear stochastic control problem and then in a fairly general formulation for the singular stochastic control problem of irreversible investment with inventory risk. For the latter, we illustrate in a first case study how different signals on exogenous jumps lead to different optimal controls, interpolating between the predictable and the optional case in a systematic manner.

Optimal reduction of public debt under partial observation of the economic growth

We consider a government that aims at reducing the debt-to-(gross domestic product) (GDP) ratio of a country. The government observes the level of the debt-to-GDP ratio and an indicator of the state of the economy, but does not directly observe the development of the underlying macroeconomic conditions. The government’s criterion is to minimise the sum of the total expected costs of holding debt and of debt reduction policies. We model this problem as a singular stochastic control problem under partial observation. The contribution of the paper is twofold. Firstly, we provide a general formulation of the model in which the level of the debt-to-GDP ratio and the value of the macroeconomic indicator evolve as a diffusion and a jump-diffusion, respectively, with coefficients depending on the regimes of the economy. The latter are described through a finite-state continuous-time Markov chain. We reduce the original problem via filtering techniques to an equivalent one with full information (the so-called separated problem), and we provide a general verification result in terms of a related optimal stopping problem under full information. Secondly, we specialise to a case study in which the economy faces only two regimes and the macroeconomic indicator has a suitable diffusive dynamics. In this setting, we provide the optimal debt reduction policy. This is given in terms of the continuous free boundary arising in an auxiliary fully two-dimensional optimal stopping problem.

Optimal dividends and capital injection under dividend restrictions

We study a singular stochastic control problem faced by the owner of an insurance company that dynamically pays dividends and raises capital in the presence of the restriction that the surplus process must be above a given dividend payout barrier in order for dividend payments to be allowed. Bankruptcy occurs if the surplus process becomes negative and there are proportional costs for capital injection. We show that one of the following strategies is optimal: (i) Pay dividends and inject capital in order to reflect the surplus process at an upper barrier and at 0, implying bankruptcy never occurs. (ii) Pay dividends in order to reflect the surplus process at an upper barrier and never inject capital—corresponding to absorption at 0—implying bankruptcy occurs the first time the surplus reaches zero. We show that if the costs of capital injection are low, then a sufficiently high dividend payout barrier will change the optimal strategy from type (i) (without bankruptcy) to type (ii) (with bankruptcy). Moreover, if the costs are high, then the optimal strategy is of type (ii) regardless of the dividend payout barrier. We also consider the possibility for the owner to choose a stopping time at which the insurance company is liquidated and the owner obtains a liquidation value. The uncontrolled surplus process is a Wiener process with drift.

Dividend policy and capital structure of a defaultable firm

AbstractDefault risk significantly affects the corporate policies of a firm. We develop a model in which a limited liability entity subject to default at an exponential random time jointly sets its dividend policy and capital structure to maximize the expected lifetime utility from consumption of risk‐averse equity investors. We give a complete characterization of the solution to the singular stochastic control problem. The optimal policy involves paying dividends to keep the ratio of firm's equity value to investors' wealth below a critical threshold. Dividend payout acts as a precautionary channel to transfer wealth from the firm to investors for mitigation of losses in the event of default. Higher the default risk, more aggressively the firm leverages and pays dividends.

A Singular Stochastic Control Problem with Interconnected Dynamics

In this paper we study a Markovian two-dimensional bounded-variation stochastic control problem whose state process consists of a diffusive mean-reverting component and of a purely controlled one. ...