Parabolic variational inequality with parameter and gradient constraints

This paper concerns a singular control problem whose value function is governed by a time-dependent HJB equation with gradient constraints. The method is to transform a two-dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies. Moreover we analyze the behaviors of the free boundary surfaces. The main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter.     

Optimal investment with transaction costs based on exponential utility function: a parabolic double obstacle problem

This paper concerns optimal investment problem with proportional transaction costs and finite time horizon based on exponential utility function. Using a partial differential equation approach, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. Numerical examples are obtained by the binomial method.     

Optimal control of the obstacle in a quasilinear elliptic variational inequality

In this paper we consider an obstacle control problem where the state satisfies a quasilinear elliptic variational inequality and the control function is the obstacle. The state is chosen to be close to the desire profile while the H² norms of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). The financial position is given by an RCLL adapted process. We first state some properties of RBSDEs with jumps when the obstacle process is RCLL only. We then prove that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of optimal stopping times is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, we investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/optimal stopping game problems. We prove that, under some hypothesis, the value function is equal to the solution of an RBSDE. We then study the existence of saddle points when the obstacle is left-upper semi-continuous along stopping times.     

Optimal Control of the Obstacle for a Parabolic Variational Inequality

An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L²(0, T; H²(Ω) ∩ H¹₀(Ω)) with ψ_t ∈ L²(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Optimal control problems on manifolds: a dynamic programming approach

Dynamic programming identifies the value function of continuous time optimal control with a solution to the Hamilton-Jacobi equation, appropriately defined. This relationship in turn leads to sufficient conditions of global optimality, which have been widely used to confirm the optimality of putative minimisers. In continuous time optimal control, the dynamic programming methodology has been used for problems with state space a vector space. However there are many problems of interest in which it is necessary to regard the state space as a manifold. This paper extends dynamic programming to cover problems in which the state space is a general finite-dimensional C^∞ manifold. It shows that, also in a manifold setting, we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous, lower bounded, generalised solution of the Hamilton-Jacobi equation. The application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum.     

Optimal expulsion and optimal confinement of a Brownian particle with a switching cost

We solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle, by controlling its drift. The player can change from one drift to another but is subject to a switching cost. In each problem, the value function is written as the solution of a free boundary problem involving second order ordinary differential equations, in which the unknown boundaries are found by applying the principle of smooth fit. For both problems, we compute the value function, we exhibit the optimal strategy and we prove its generic uniqueness.     

Optimal control of unilateral obstacle problem with a source term

We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function which is the control function is assumed in H². We use an approximate technique to introduce a family of problems governed by variational equations. We prove optimal solutions existence and give necessary optimality conditions. The author is grateful to Prof. M. Bergounioux for her instructive suggestions.     

拟线性椭圆变分不等式的障碍最优控制问题

对拟线性椭圆变分不等式的障碍最优控制问题(即以障碍为控制变量)进行了研究．指标泛函为Lagrange型,其中含有控制变量二阶导数的p次幂,这使得最优性条件的推导颇为不易．对所考虑的问题给出了最优控制的存在性定理以及必要条件．     

A computational method for free terminal time optimal control problem governed by nonlinear time delayed systems

This paper considers a free terminal time optimal control problem governed by nonlinear time delayed system, where both the terminal time and the control are required to be determined such that a cost function is minimized subject to continuous inequality state constraints. To solve this free terminal time optimal control problem, the control parameterization technique is applied to approximate the control function as a piecewise constant control function, where both the heights and the switching times are regarded as decision variables. In this way, the free terminal time optimal control problem is approximated as a sequence of optimal parameter selection problems governed by nonlinear time delayed systems, each of which can be viewed as a nonlinear optimization problem. Then, a fully informed particle swarm optimization method is adopted to solve the approximate problem. Finally, two free terminal time optimal control problems, including an optimal fishery control problem, are solved by using the proposed method so as to demonstrate its applicability.     

Numerical solution of an optimal investment problem with proportional transaction costs

This paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite-horizon optimal investment problem with proportional transaction costs. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable. The proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain, a characteristics method for time discretization to deal with the large gradients of the solution, a Newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle (free boundary) feature. Moreover, piecewise linear Lagrange finite elements for spatial discretization are considered. Numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in Dai and Yi (2009) [6].     

Porosity of free boundaries in the obstacle problem for quasilinear elliptic equations

In this paper, we establish growth rate of solutions near free boundaries in the identical zero obstacle problem for quasilinear elliptic equations. As a result, we obtain porosity of free boundaries, which is naturally an extension of the previous works by Karp et al. (J. Diff. Equ. 164 (2000) 110–117) for p-Laplacian equations, and by Zheng and Zhang (J. Shaanxi Normal Univ. 40(2) (2012) 11–13, 18) for p-Laplacian type equations.     

Optimal control of the obstacle in semilinear variational inequalities

We consider an optimal control problem where the state satisfies an obstacle type semilinear variational inequality and the control function is the obstacle. The state is chosen to be close to a desired profile while the obstacle is not too large in H ₀ ¹ (), and H ²-bounded. We prove that an optimal control exists and give necessary optimality conditions, using approximation techniques.     

Regularity properties in a state-constrained expected utility maximization problem

We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of an optimal strategy under rather mild model assumptions. This will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove a dynamic programming principle without appealing to the classical measurable selection arguments. This permits us to establish a tight relation between our value function and a nonlinear parabolic degenerated Hamilton–Jacobi–Bellman (HJB) equation with singularity. To conclude, we show a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.     

OPTIMAL RESOURCE DEVELOPMENT AND IRREVERSIBILITIES: COOPERATIVE AND NONCOOPERATIVE SOLUTIONS

This paper derives and compares free boundaries for the problem of irreversible development of an environmental resource under uncertainty, by explicitly taking into account the facts that: (i) there might be more than one private profit-maximizer decision-maker that acquires profits by developing the resource involved in the problem; (ii) there might be interactions among these decision-makers, in the sense that the development undertaken by a certain developer might affect the cost of the rest; and (iii) the undeveloped resource has an environmental value which is not taken into account by the individual developers but might be accounted for in the context of an optimal development problem faced by a social planner or environmental regulator. By comparing the three resulting free boundaries, it is shown that the noncooperative solution implies the fastest development as compared to cooperative outcomes. Policy schemes in the form of development fees and development limits which can secure the cooperative outcome are determined.     

Deterministic Time-inconsistent Optimal Control Problems - an Essentially Cooperative Approach

A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N -person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.     

Optimal stochastic control of the intensity of point processes

We consider an intensity control problem for a point process to maximize the expectation of a function of the time when the nth event occurs. We find the optimal control policy when the objective function is unimodal. Moreover, if the objective function is log-concave, so is the value function. As an application, we completely solve an intensity control problem that generalizes the problem studied by Brémaud (1976) and Defourny (2018). Also, we resolve the two conjectures made by Defourny (2018).     

Uniformly Lipschitz feedback optimal controls in a linear-quadratic framework

A uniformly k-Lipschitz feedback optimal control problem is considered in a linear quadratic framework. The value function is derived by a comparison-based reasoning, via which a necessary condition to the existence of optimal solutions is obtained: optimal feedback controls must be linear.     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

A stochastic partially reversible investment problem on a finite time-horizon: Free-boundary analysis

We study a continuous-time, finite horizon, stochastic partially reversible investment problem for a firm producing a single good in a market with frictions. The production capacity is modeled as a one-dimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment–disinvestment strategy. We associate to the investment–disinvestment problem a zero-sum optimal stopping game and characterize its value function through a free-boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment–disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.