Stochastic Impulse Control Problem Research Articles

The solution to an impulse control problem motivated by optimal harvesting

We consider a stochastic impulse control problem that is motivated by applications such as the optimal exploitation of a natural resource. In particular, we consider a stochastic system whose uncontrolled state dynamics are modelled by a non-explosive positive linear diffusion. The control that can be applied to this system takes the form of one-sided impulsive action. The objective of the control problem is to maximise a discounted performance criterion that rewards the effect of control action but involves a fixed cost at each time of a control intervention. We derive the complete solution to this problem under general assumptions. It turns out that the solution can take four qualitatively different forms, several of which have not been observed in the literature. In two of the four cases, there exist only ε-optimal control strategies. We also show that the boundary classification of 0 may play a critical role in the solution of the problem. Furthermore, we develop a way for establishing the strong solution to a stochastic impulse control problem's optimally controlled SDE.

Optimal stopping and impulse control in the presence of an anticipated regime switch

We consider a class of stochastic optimal stopping and impulse control problems where the agent solving the problem anticipates that a regime switch will happen at a random time in the future. We assume that there are only two regimes, the regime switching time is exponentially distributed, the underlying stochastic process is a linear, regular, time-homogeneous diffusion in both regimes and the payoff may be regime-dependent. This is in contrast with most existing literature on the topic, where regime switching is modulated by a continuous-time Markov chain and the underlying process and payoff belong to the same parametric family in all regimes. We state a set of easily verifiable sufficient conditions under which the solutions to these problems are given by one-sided threshold strategies. We prove uniqueness of the thresholds and characterize them as solutions to certain algebraic equations. We also study how anticipation affects optimal policies i.e. we present various comparison results for problems with and without regime switching. It may happen that the anticipative value functions and optimal policies coincide with the usual ones even if the regime switching structure is non-trivial. We illustrate our results with practical examples.

Moment-constrained optimal dividends: precommitment and consistent planning

AbstractA moment constraint that limits the number of dividends in an optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz’s consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame-perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions

A stochastic impulse control problem with imperfect controllability of interventions is formulated with an emphasis on applications to ecological and environmental management problems. The imperfectness comes from uncertainties with respect to the magnitude of interventions. Our model is based on a dynamic programming formalism to impulsively control a 1-D diffusion process of a geometric Brownian type. The imperfectness leads to a non-local operator different from the many conventional ones, and evokes a slightly different optimal intervention policy. We give viscosity characterizations of the Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) governing the value function focusing on its numerical computation. Uniqueness and verification results of the HJBQVI are presented and a candidate exact solution is constructed. The HJBQVI is solved with the two different numerical methods, an ordinary differential equation (ODE) based method and a finite difference scheme, demonstrating their consistency. Furthermore, the resulting controlled dynamics are extensively analyzed focusing on a bird population management case from a statistical standpoint.

Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem

AbstractNonsmooth dynamics driven by stochastic disturbance arise in a wide variety of engineering problems. Impulsive interventions are often employed to control stochastic systems; however, the modeling and analysis subject to execution delay have been less explored. In addition, continuously receiving information of the dynamics is not always possible. In this article, with an application to an environmental restoration problem, a continuous‐time stochastic impulse control problem subject to execution delay under discrete and random observations is newly formulated and analyzed. The dynamics have a nonsmooth coefficient modulated by a Markov chain, and eventually attain an undesirable state like a depletion due to the nonsmoothness. The goal of the control problem is to find the most cost‐efficient policy that can prevent the dynamics from attaining the undesirable state. We demonstrate that finding the optimal policy reduces to solving a nonstandard system of degenerate elliptic equations, the Hamilton–Jacobi–Bellman equation (HJBE), which is rigorously and analytically verified in a simplified case. The associated Fokker–Planck equation (FPE) for the controlled dynamics is derived and solved explicitly as well. The model is finally applied to numerical computation of a recent river environmental restoration problem. The HJBE and FPE are successfully computed, and the optimal policy and the probability density functions are numerically obtained. The impacts of execution delay are discussed to deeper analyze the model.

Stochastic impulse control Problem with state and time dependent cost functions

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Effect of Zero-cost Interval in Running Cost for Stochastic Impulse Control Problems

Effect of Zero-cost Interval in Running Cost for Stochastic Impulse Control Problems

Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates

We consider a retailer who buys energy in the wholesale market and resells it to final consumers. The retailer has to decide when to intervene to change the price he asks to his customers, in order to maximize his income. We model the problem as an infinite-horizon stochastic impulse control problem. We characterize an optimal price strategy and provide analytical existence results for the equations involved. We then investigate the dependence on the intervention cost. In particular, we prove that the measure of the continuation region is asymptotic to the fourth root of the cost. Finally, we provide some numerical results and consider a suitable extension of the model.

Super-replication with fixed transaction costs

We study super-replication of contingent claims in markets with fixed transaction costs. This can be viewed as a stochastic impulse control problem with a terminal state constraint. The first result in this paper reveals that in reasonable continuous time financial market models the super-replication price is prohibitively costly and leads to trivial buy-and-hold strategies. Our second result derives nontrivial scaling limits of super-replication prices for binomial models with small fixed costs.

Irreversible investment with fixed adjustment costs: a stochastic impulse control approach

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.

Optimal stochastic impulse and regular control for capital injections: A hybrid strategy

This paper investigates an optimal capital injection problem for a large company whose cash flow is described by a drifted Brownian motion. We firstly bring forward the fact that the decision maker can choose the types of capital injection including regular or/and impulse injections in practice. The decision maker has the option to decide the capital injection rate (with a maximum restriction) of regular injection and the capital injection time and the amount of impulse injection as well. We assume that both types of capital injections have proportional costs, and an additional fixed cost occurs associated with each impulse capital injection. From the point of view of cost control, the purpose of the decision maker is to find a strategy to minimize the cost of capital injections, with the minimal requirement that the firm has a positive cash flow. It leads to a mixed stochastic impulse and regular control problem. Under this criterion, we obtain the value function and the optimal control strategy which is a hybrid type of impulse and regular capital injections. At last, we also show the sensibilities of model parameters to the optimal strategy and the value function.

Stochastic impulse control with regime-switching dynamics

Optimal product management problems with multiple product generations in continuous time lead to the consideration of dynamic optimal control problems that feature both intervention costs and partially controlled regime shifts. We therefore investigate and solve such stochastic impulse control problems with regime-switching in a general setting. We analyze the associated coupled systems of quasi-variational inequalities in suitable Sobolev spaces, and we establish a direct approach to construct both the value function and optimal strategies. Our results in particular yield a numerical method for the computation of the optimal value function and the associated strategies.

A General Verification Result for Stochastic Impulse Control Problems

A General Verification Result for Stochastic Impulse Control Problems

随机递归系统的脉冲控制的近似最大值原理

Here, we discuss the near-optimality for a class of stochastic impulse control problems. The state process in our problem is given by forward-backward stochastic differential equations (FBSDEs) with two control components involved: the regular and impulse control. More specially, the impulse control is defined on a sequence of prescribed stopping times. A recursive cost functional is introduced and the maximum principle for its near-optimality (both necessary and sufficient conditions) is derived with the help of Ekeland’s principle and variational analysis. For illustration, one concrete example is studied with our maximum principle and the corresponding near-optimal control is characterized.

A General Verification Result for Stochastic Impulse Control Problems

This paper establishes existence of optimal controls for a general stochastic impulse control problem. For this, the value function is characterized as the pointwise minimum of a set of superharmonic functions, as the unique continuous viscosity solution of the quasi-variational inequalities (QVIs), and as the limit of a sequence of iterated optimal stopping problems. A combination of these characterizations is used to construct optimal controls without relying on any regularity of the value function beyond continuity. Our approach is based on the stochastic Perron method and the assumption that the associated QVIs satisfy a comparison principle.

A Maximum Principle via Malliavin calculus for combined stochastic control and impulse control of forward‐backward systems

AbstractWe consider a combined stochastic control and impulse control problem of forward‐backward systems driven by Lévy processes, where both the system coefficients and the objective performance functional are allowed to be random, non‐Markovian; the information available to the controller is partial information. Applying a Malliavin calculus approach, we derive a maximum principle for this control problem, where the adjoint processes are explicitly represented by the parameters and the states of the system. Finally, we give two examples of applications. © 2015 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

Some Numerical Methods for Solving Stochastic Impulse Control in Natural Gas Storage Facilities

The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP).

Stochastic impulse control with regime switching for the optimal dividend policy when there are business cycles, taxes and fixed costs

We consider the problem of finding the optimal dividend policy for a company whose cash reserve follows a Brownian motion with drift and volatility modulated by an observable finite-state continuous-time Markov chain. The Markov chain represents the regime of the economy. We allow fixed costs and taxes associated with the dividend payments. This optimization problem generates a stochastic impulse control problem with regime switching. We solve this problem and obtain the first analytical solutions for the optimal dividend policy when there are simultaneously fixed costs, taxes and business cycles. Our results show that the optimal dividend policy depends strongly on the regime of the economy, on fixed costs and on taxes.

Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes

In this paper we study the optimal excess-of-loss reinsurance and dividend strategy for maximizing the expected total discounted dividends received by shareholders until ruin time. Transaction costs and taxes are required when dividends occur. The problem is formulated as a stochastic impulse control problem. By solving the corresponding quasi-variational inequality, we obtain analytical solutions for the optimal return function and the optimal strategy.

Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes

In this paper, we study optimal dividend problem in the classical risk model. Transaction costs and taxes are required when dividends occur. The problem is formulated as a stochastic impulse control problem. By solving the corresponding quasi-variational inequality, we obtain the analytical solutions of the optimal return function and the optimal dividend strategy when claims are exponentially distributed. We also find a formula for the expected time between dividends. The results show that, as the dividend tax rate decreases, it is optimal for the shareholders to receive smaller but more frequent dividend payments.