Constrained Viscosity Solution Research Articles

Nonzero-Sum Stochastic Impulse Games with an Application in Competitive Retail Energy Markets

We study a nonzero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The objective of each player is to maximize her total expected discounted profits. The resolution methodology relies on the connection between Nash equilibrium and the corresponding system of quasi-variational inequalities (QVIs in short). We prove, by means of the weak dynamic programming principle for the stochastic differential game, that the equilibrium expected payoff of each player is a constrained viscosity solution to the associated QVIs system in the class of linear growth functions. We also introduce a family of equilibrium expected payoffs converging to our equilibrium expected payoff of each player, and which is characterized as the unique constrained viscosity solutions of an approximation of our QVIs system. This convergence result is useful for numerical purpose. We apply a probabilistic numerical scheme which approximates the solution of the QVIs system to the case of the competition between two electricity retailers. We show how our model reproduces the qualitative behavior of electricity retail competition.

Regime-switching constrained viscosity solutions approach for controlling dam–reservoir systems

A new stochastic control problem of a dam–reservoir system installed in a river is analyzed both mathematically and numerically. Water balance dynamics of the reservoir are piece-wise deterministic and are driven by a stochastic regime-switching inflow process. The system is controlled to balance among the operation purpose and the internal and downstream environmental conditions. Finding the optimal operation policy of the system reduces to solving an optimality equation with a discontinuous Hamiltonian, which is a system of nonlinear degenerate parabolic (or hyperbolic) equations. We show that the optimality equation has at most one constrained viscosity solution and find the solution explicitly under certain conditions. The model is applied to numerical computation of the operation policy of an existing dam–reservoir system using a high-order finite difference scheme. The computational results can suggest how the operation policy should be adapted according to the environmental concerns of the river.

Optimal Singular Dividend Problem Under the Sparre Andersen Model

In this paper, we study the optimal dividend problem assuming that the underlying reserve process follows the Sparre Andersen model. In this model, there is no constant restriction on the dividend rates, i.e., the optimization problem is of singular type. In this case, the value function is no longer bounded and the associated Hamilton–Jacobi–Bellman equation is a variational inequality involving a first-order integro-differential operator and a gradient constraint. We prove the regularity properties for the value function by constructing strategies and show that the value function is a constrained viscosity solution of the associated Hamilton–Jacobi–Bellman equation. In addition, we prove that the value function is the upper semicontinuous envelope of the supremum for a class of subsolutions.

Explicit solutions to utility maximization problems in a regime-switching market model via Laplace transforms

We study the problem of utility maximization from terminal wealth in which an agent optimally builds her portfolio by investing in a bond and a risky asset. The asset price dynamics follow a diffusion process with regime-switching coefficients modeled by a continuous-time finite-state Markov chain. We consider an investor with a Constant Relative Risk Aversion (CRRA) utility function. We deduce the associated Hamilton–Jacobi–Bellman equation to construct the solution and the optimal trading strategy and verify optimality by showing that the value function is the unique constrained viscosity solution of the HJB equation. By means of a Laplace transform method, we show how to explicitly compute the value function and illustrate the method with the two- and three-states cases. This method is interesting in its own right and can be adapted in other applications involving hybrid systems and using other types of transforms with basic properties similar to the Laplace transform.

Optimal Control of Continuous-Time Markov Chains with Noise-Free Observation

We consider an infinite horizon optimal control problem for a continuous-time Markov chain $X$ in a finite set $I$ with noise-free partial observation. The observation process is defined as $Y_t = h(X_t)$, $t \geq 0$, where $h$ is a given map defined on $I$. The observation is noise-free in the sense that the only source of randomness is the process $X$ itself. The aim is to minimize a discounted cost functional and study the associated value function $V$. After transforming the control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we provide a link between the value function $v$ associated to the latter control problem and the original value function $V$. Then, we present two different characterizations of $v$ (and indirectly of $V$): on one hand as the unique fixed point of a suitably defined contraction mapping and on the other hand as the unique constrained viscosity solution (in the sense of Soner) of a HJB integro-differential equation. Under suitable assumptions, we finally prove the existence of an optimal control.

Optimal dividend and investment problems under Sparre Andersen model

In this paper, we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a “renewal” process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique, we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution to the associated HJB equation on a cylindrical domain on which the problem is well defined.

Hamilton–Jacobi–Bellman equations for optimal control processes with convex state constraints

This work aims at studying some optimal control problems with convex state constraint sets. It is known that for state constrained problems, and when the state constraint set coincides with the closure of its interior, the value function satisfies a Hamilton–Jacobi equation in the constrained viscosity sense. This notion of solution has been introduced by H.M. Soner (1986) and provides a characterization of the value functions in many situations where an inward pointing condition (IPC) is satisfied. Here, we first identify a class of control problems where the constrained viscosity notion is still suitable to characterize the value function without requiring the IPC. Moreover, we generalize the notion of constrained viscosity solutions to some situations where the state constraint set has an empty interior.

On singular control problems with state constraints and regime-switching: A viscosity solution approach

This paper investigates a singular stochastic control problem for a multi-dimensional regime-switching diffusion process confined in an unbounded domain. The objective is to maximize the total expected discounted rewards from exerting the singular control. Such a formulation stems from application areas such as optimal harvesting multiple species and optimal dividends payments schemes in random environments. With the aid of weak dynamic programming principle and an exponential transformation, we characterize the value function to be the unique constrained viscosity solution of a certain system of coupled nonlinear quasi-variational inequalities. Several examples are analyzed in detail to demonstrate the main results.

Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing

We consider the valuation of a block of perpetual ESOs and the optimalexercise decision for an employee endowed with them and with trading restrictions.A fluid model is proposed to characterize the exercise process.The objective is to maximize the overall discountreturns for the employee through exercising the options over time.The optimal value function is defined as the grant-date fair value of the block of options, and is thenshown by the dynamic programming principle to be a continuous constrained viscosity solutionto the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully nonlinear second order elliptic partial differential equation (PDE)in the plane.We prove the comparison principle and the uniqueness.The numerical simulation is discussed and the corresponding optimal decision turns out to be a threshold-style strategy. These results provide an appropriate method to estimate the cost of the ESOs for the company andalso offer favorable suggestions on selecting right momentsto exercise the options over time for the employee.

Infinite horizon problems on stratifiable state-constraints sets

This paper deals with a state-constrained control problem. It is well known that, unless some compatibility condition between constraints and dynamics holds, the Value Function has not enough regularity, or can fail to be the unique constrained viscosity solution of a Hamilton–Jacobi–Bellman (HJB) equation. Here, we consider the case of a set of constraints having a stratified structure. Under this circumstance, the interior of this set may be empty or disconnected, and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints. In such situations, the classical pointing qualification hypothesis is not relevant. The discontinuous Value Function is then characterized by means of a system of HJB equations on each stratum that composes the state-constraints. This result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur.

Stochastic Perron's method for optimal control problems with state constraints

We apply the stochastic Perron method of Bayraktar and Sîrbu to a general infinite horizon optimal control problem, where the state $X$ is a controlled diffusion process, and the state constraint is described by a closed set. We prove that the value function $v$ is bounded from below (resp., from above) by a viscosity supersolution (resp., subsolution) of the related state constrained problem for the Hamilton-Jacobi-Bellman equation. In the case of a smooth domain, under some additional assumptions, these estimates allow to identify $v$ with a unique continuous constrained viscosity solution of this equation.

Numerical methods for an optimal order execution problem

This paper deals with numerical solutions to an impulse control problem arising from optimal portfolio liquidation with bid-ask spread and market price impact pena\-lizing speedy execution trades. The corresponding dynamic programming (DP) equation is a quasi-variational inequality (QVI) with solvency constraint satisfied by the value function in the sense of constrained viscosity solutions. By taking advantage of the lag variable tracking the time interval between trades, we can provide an explicit backward numerical scheme for the time discretization of the DPQVI. The convergence of this discrete-time scheme is shown by viscosity solutions arguments. An optimal quantization method is used for computing the (conditional) expectations arising in this scheme. Numerical results are presented by examining the behaviour of optimal liquidation strategies, and comparative performance analysis with respect to some benchmark execution strategies. We also illustrate our optimal liquidation algorithm on real data, and observe various interesting patterns of order execution strategies. Finally, we provide some numerical tests of sensitivity with respect to the bid/ask spread and market impact parameters.

Optimal portfolio and consumption selection with default risk

We investigate an optimal portfolio and consumption choice problem with a defaultable security. Under the goal of maximizing the expected discounted utility of the average past consumption, a dynamic programming principle is applied to derive a pair of second-order parabolic Hamilton-Jacobi-Bellman (HJB) equations with gradient constraints. We explore these HJB equations by a viscosity solution approach and characterize the post-default and pre-default value functions as a unique pair of constrained viscosity solutions to the HJB equations.

Hamilton–Jacobi equations constrained on networks

We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. In our setting, the value function is continuous. We de ne a notion of constrained viscosity solution of Hamilton-Jacobi equations on the network and we study related comparison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained viscosity solution of the Hamilton-Jacobi equation on the network.

Optimal Decision for Selling an Illiquid Stock

This paper is concerned with liquidation of an illiquid stock. The stock price follows a fluid model which is dictated by the rates of selling and buying over time. The objective is to maximize the expected overall return. The method of constrained viscosity solution is used to characterize the dynamics governing the optimal reward function and the associated boundary conditions. Numerical examples are given to illustrate the results.

Portfolio Optimization without the Self-Financing Assumption

In this paper, we relax the assumption of a self-financing strategy in the dynamic investment models. In so doing we provide smooth solutions and constrained viscosity solutions.

Optimal control problems of BV trajectories with pointwise state constraints

AbstractThis paper deals with some optimal control problems governed by ordinary differential equations with Radon measures as data and with pointwise state constraints. We study the properties of the value function and obtain its characterization by means of an auxiliary control problem of absolutely continuous trajectories. For this, we use some known techniques of reparametrization and graph completion.We are also interested in the characterization of the value function as the unique constrained viscosity solution of a Hamilton-Jacobi equation with measurable time dependant Hamiltonians.

Numerical Methods for an Optimal Order Execution Problem

This paper deals with numerical solutions to an impulse control problem arising from optimal portfolio liquidation with bid-ask spread and market price impact penalizing speedy execution trades. The corresponding dynamic programming (DP) equation is a quasi-variational inequality QVI) with solvency constraint satisfied by the value function in the sense of constrained viscosity solutions. By taking advantage of the lag variable tracking the time interval between trades, we can provide an explicit backward numerical scheme for the time discretization of the DPQVI. The convergence of this discrete-time scheme is shown by viscosity solutions arguments. An optimal quantization method is used for computing the (conditional) expectations arising in this scheme. Numerical results are presented by examining the behaviour of optimal liquidation strategies, and comparative performance analysis with respect to some benchmark execution strategies. We also illustrate our optimal liquidation algorithm on real data, and observe various interesting patterns of order execution strategies. Finally, we provide some numerical tests of sensitivity with respect to the bid/ask spread and market impact parameters.

Portfolio Optimization Without the Self-Financing Assumption

In this paper, we relax the assumption of a self-financing strategy in the dynamic investment models. In so doing we provide smooth solutions and constrained viscosity solutions.

Optimal Portfolio Liquidation with Execution Cost and Risk

We study the optimal portfolio liquidation problem over a finite horizon in a limit order book with bid-ask spread and temporary market price impact penalizing speedy execution trades. We use a continuous-time modeling framework, but in contrast with previous related papers (see, e.g., [L. C. G. Rogers and S. Singh, Math. Finance, 20 (2010), pp. 597–615] and [A. Schied and T. Schöneborn, Finance Stoch., 13 (2009), pp. 181–204]), we do not assume continuous-time trading strategies. We consider instead real trading that occur in discrete time, and this is formulated as an impulse control problem under a solvency constraint, including the lag variable tracking the time interval between trades. A first important result of our paper is to prove rigorously that nearly optimal execution strategies in this context actually lead to a finite number of trades with strictly increasing trading times, and this holds true without assuming ad hoc any fixed transaction fee. Next, we derive the dynamic programming quasi-variational inequality satisfied by the value function in the sense of constrained viscosity solutions. We also introduce a family of value functions which converges to our value function and is characterized as the unique constrained viscosity solutions of an approximation of our dynamic programming equation. This convergence result is useful for numerical purpose but is postponed until a companion paper [F. Guilbaud, M. Mnif, and H. Pham, Numerical Methods for an Optimal Order Execution Problem, preprint, 2010].