Viscosity Solution Approach Research Articles

The two membranes problem for different operators

We study the two membranes problem for different operators, possibly nonlocal. We prove a general result about the Hölder continuity of the solutions and we develop a viscosity solution approach to this problem. Then we obtain C1,γ regularity of the solutions provided that the orders of the two operators are different. In the special case when one operator coincides with the fractional Laplacian, we obtain the optimal regularity and a characterization of the free boundary.

Liquidity risk and optimal dividend/investment strategies

In this paper, we study the problem of determining an optimal control on the dividend and investment policy of a firm operating under uncertain environment and risk constraints. We allow the company to make investment decisions by acquiring or selling producing assets whose value is governed by a stochastic process. The firm may face liquidity costs when it decides to buy or sell assets. We formulate this problem as a multi-dimensional mixed singular and multi-switching control problem and use a viscosity solution approach. We numerically compute our optimal strategies and enrich our studies with numerical results and illustrations.

Weak solutions for a stochastic mean curvature flow of two-dimensional graphs

We study a stochastically perturbed mean curvature flow for graphs in $${\mathbb {R}}^3$$ over the two-dimensional unit-cube subject to periodic boundary conditions. The stochastic perturbation is a one dimensional white noise acting uniformly in all points of the surface in normal direction. We establish the existence of a weak martingale solution. The proof is based on energy methods and therefore presents an alternative to the stochastic viscosity solution approach. To overcome difficulties induced by the degeneracy of the mean curvature operator and the multiplicative gradient noise present in the model we employ a three step approximation scheme together with refined stochastic compactness and martingale identification methods.

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.

Optimal switching for the pairs trading rule: A viscosity solutions approach

This paper studies the problem of determining the optimal cut-off for pairs trading rules. We consider two cointegrated assets whose spread is modelled by a general mean-reverting process, and the optimal pair trading rule is formulated as an optimal switching problem between three regimes: flat position (no holding stocks), one long and the other short and one short and the other long. A fixed commission cost is charged with each transaction. We use a viscosity solutions approach to prove the existence and the explicit characterisation of cut-off points via the resolution of quasi-algebraic equations. We illustrate our results by numerical simulations.

Free boundary problems for tumor growth: A viscosity solutions approach

The mathematical modeling of tumor growth leads to singular “stiff pressure law” limits for porous medium equations with a source term. Such asymptotic problems give rise to free boundaries, which, in the absence of active motion, are generalized Hele-Shaw flows. In this note we use viscosity solutions methods to study limits for porous medium-type equations with active motion. We prove the uniform convergence of the density under fairly general assumptions on the initial data, thus improving existing results. We also obtain some additional information/regularity about the propagating interfaces, which, in view of the discontinuities, can nucleate and, thus, change topological type. The main tool is the construction of local, smooth, radial solutions which serve as barriers for the existence and uniqueness results as well as to quantify the speed of propagation of the free boundary propagation.

Optimal switching strategy of a mean-reverting asset over multiple regimes

We solve optimal iterative three-regime switching problems with transaction costs, with investment in a mean-reverting asset that follows an Ornstein–Uhlenbeck process and find the explicit solutions. The investor can take either a long, short or square position and can switch positions during the period. Modeling the short sales position is necessary to study optimal trading strategies such as the pair trading. Few studies provide explicit solutions to problems with multiple (more than two) regimes (states). The value function is proved to be a unique viscosity solution of a Hamilton–Jacobi–Bellman variational inequality (HJB-VI). Multiple-regime switching problems are more difficult to solve than conventional two-regime switching problems, because they need to identify not only when to switch, but also where to switch. Therefore, multiple-regime switching problems need to identify the structure of the continuation/switching regions in the free boundary problem for each regime. If the number of the states N is two, only two regions have to be identified, but if N=3, NP2=6 regions have to be detected. We identify the structure of the switching regions for each regime using the theories related to the viscosity solution approach.

Characterization of the Optimal Boundaries in Reversible Investment Problems

This paper studies a reversible investment problem where a social planner aims to control its capacity production in order to fit optimally the random demand of a good. Our model allows for general diffusion dynamics on the demand as well as general cost functional. The resulting optimization problem leads to a degenerate two-dimensional bounded variation singular stochastic control problem, for which explicit solution is not available in general and the standard verification approach cannot be applied a priori. We use a direct viscosity solutions approach for deriving some features of the optimal free boundary function and for displaying the structure of the solution. In the quadratic cost case, we are able to prove a smooth fit $C^2$ property, which gives rise to a full characterization of the optimal boundaries and value function.

Large Deviations Principle by viscosity solutions: The case of diffusions with oblique Lipschitz reflections

Nous etablissons un principe de Grandes Deviations pour des diffusions reflechies obliquement sur le bord d’un domaine regulier lorsque la direction de la reflection est Lipschitz. La fonction de taux s’exprime comme la fonction valeur d’un probleme d’arret optimal et est compacte. Nous utilisons des techniques de solutions de viscosite. Les probabilites recherchees sont interpretees comme des solutions de certaines EDPs, leur transformees logarithmiques donnent lieu a de nouvelles equations dans lesquelles il est aise de passer a la limites. Enfin les fonctionnelles d’action sont identifiees comme etant les solutions des dites equations limite.

An Optimal Dividend and Investment Control Problem under Debt Constraints

This paper concerns the problem of determining an optimal control on the dividend and investment policy of a firm under debt constraints. We allow the company to make investment by increasing its outstanding indebtedness, which would impact its capital structure and risk profile, thus resulting in higher interest rate debts. Moreover, a high level of debt is also a challenging constraint to any firm, as it is the threshold below which the firm value should never go to avoid bankruptcy. It is equally possible for the firm to divest parts of its business in order to decrease its financial debt owed to creditors. In addition, the firm may favor investment by postponing or reducing any dividend distribution to shareholders. We formulate this problem as a combined singular and multiswitching control problem and use a viscosity solution approach to get qualitative descriptions of the solution. We further enrich our studies with a complete resolution of the problem in the two-regime case and provide some numeric...

Stochastic optimal multi-modes switching with a viscosity solution approach

We consider the problem of optimal multi-modes switching in finite horizon, when the state of the system, including the switching cost functions are arbitrary (gij(t,x)≥0). We show existence of the optimal strategy, via a verification theorem. Finally, when the state of the system is a Markov process, we show that the vector of value functions of the optimal problem is the unique viscosity solution to the system of m variational partial differential inequalities with inter-connected obstacles.

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.

OPTIMAL MULTI-MODES SWITCHING PROBLEM IN INFINITE HORIZON

This paper studies the problem of the deterministic version of the Verification Theorem for the optimal m-states switching in infinite horizon under Markovian framework with arbitrary switching cost functions. The problem is formulated as an extended impulse control problem and solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. A viscosity solution approach is employed to carry out a fine analysis on the associated system of m variational inequalities with inter-connected obstacles. We show that the vector of value functions of the optimal problem is the unique viscosity solution to the system. This problem is in relation with the valuation of firms in a financial market.

Adjoint and Compensated Compactness Methods for Hamilton–Jacobi PDE

We investigate the vanishing viscosity limit for Hamilton–Jacobi PDE with nonconvex Hamiltonians, and present a new method to augment the standard viscosity solution approach. The main idea is to introduce a solution σ e of the adjoint of the formal linearization, and then to integrate by parts with respect to the density σ e . This procedure leads to a natural phase space kinetic formulation and also to a new compensated compactness technique.

A Necessary condition on comparison theorem for one-dimensional stochastic differential equation

We consider the comparison theorem of one-dimensional stochastic differential equation with non-Lipschitz diffusion coefficient. Considering the two one-dimensional stochastic differential equations as a two-dimensional equation, we present a necessary condition such that comparison theorem holds by viscosity solution approach.

Optimal Switching over Multiple Regimes

This paper studies the optimal switching problem for a general one-dimensional diffusion with multiple (more than two) regimes. This is motivated in the real options literature by the investment problem of a firm managing several production modes while facing uncertainties. A viscosity solutions approach is employed to carry out a fine analysis on the associated system of variational inequalities, leading to sharp qualitative characterizations of the switching regions. These characterizations, in turn, reduce the switching problem into one of finding a finite number of threshold values in a state that would trigger switchings. The results of our analysis take several qualitatively different forms depending on model parameters, and the issue of when and where it is optimal to switch is addressed. The general results are then demonstrated by the three-regime case, where a quasi-explicit solution is obtained, and a numerical procedure to find these critical values is devised in terms of the expectation functionals of hitting times for one-dimensional diffusions.

Smooth Fit Principle for Impulse Control of Multidimensional Diffusion Processes

Value functions of impulse control problems are known to satisfy quasi-variational inequalities (QVIs) [A. Bensoussan and J.-L. Lions, Impulse Control and Quasivariational Inequalities, Heyden & Son, Philadelphia, 1984; translation of Contrôle Impulsionnel et Inéquations Quasi Variationnelles, Gauthier-Villars, Paris, 1982]. This paper proves the smooth-fit $C^1$ property of the value function for multidimensional controlled diffusions, using a viscosity solution approach. We show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions.

Finite Time Merton Strategy under Drawdown Constraint: A Viscosity Solution Approach

We consider the optimal consumption-investment problem under the drawdown constraint, i.e. the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the constant coefficients Black and Scholes model and we consider a general class of utility functions. On an infinite time horizon, Elie and Touzi (Preprint, [2006]) provided the value function as well as the optimal consumption and investment strategy in explicit form. In a more realistic setting, we consider here an agent optimizing its consumption-investment strategy on a finite time horizon. The value function interprets as the unique discontinuous viscosity solution of its corresponding Hamilton-Jacobi-Bellman equation. This leads to a numerical approximation of the value function and allows for a comparison with the explicit solution in infinite horizon.

A Viscosity Solution Approach to the Infinite-Dimensional HJB Equation Related to a Boundary Control Problem in a Transport Equation

The paper concerns the infinite-dimensional Hamilton–Jacobi–Bellman equation related to an optimal control problem regulated by a linear transport equation with boundary control. A suitable viscosity solution approach is needed in view of the presence of the unbounded control-related term in the state equation in the Hilbert setting. An existence-and-uniqueness result is obtained.

Switching Games of Stochastic Differential Systems

A two-player, zero-sum, switching game is formulated for general stochastic differential systems and is studied using a combined dynamic programming and viscosity solution approach. The existence of the game value is proved. For the proof of the related dynamic programming principle (DDP) for the lower and upper value functions, the measurability problem, of the same kind as mentioned in the paper of Fleming and Souganidis, is also encountered, and we are able to get around it via a delicate adaptation of their technique. Moreover, the traditional direct method to prove the time continuity of lower and upper value functions also gives rise to a serious measurability problem. To get around the new difficulty, a subtle dynamic programming argument is developed to obtain the time continuity, which in return is used to derive the DDP for random intermediate times from the DDP with deterministic intermediate times.