Discontinuous Viscosity Solutions Research Articles

Forward dynamic utility functions: A new model and new results

A major obstacle in the existing models of forward dynamic utilities and investment performance evaluation is to establish the existence and uniqueness of the optimal solutions. Consequently, we present a new model of forward dynamic utilities. In doing so, we establish the existence and uniqueness of the solutions for a general (smooth) utility function, and we show that the assumptions needed for such solutions are similar to those under the backward formulation. Moreover, we provide unique viscosity solutions. We also provide discontinuous viscosity solutions. In addition, we introduce Hausdorff-continuous viscosity solutions to the portfolio model.

Optimal Control of Quantum Spin Systems with Non-Zero Impulsive Control Costs

AbstractThis article introduces a generalization of problems in the domain of the optimal control of quantum systems viz. costs imposed on impulsive and bounded controls for quantum spin systems with drift. The dynamic programming approach from optimal control is used to analyze the former, and the discontinuous nature of the optimal cost function is used to motivate the need for an interpretation in terms of discontinuous viscosity solutions. Approximate solutions to example problems on a one qbit system are obtained using a value iteration approach in order to observe the effect of changes in the cost of impulses on the associated optimal cost function.

An Efficient Method for Multiobjective Optimal Control and Optimal Control Subject to Integral Constraints

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.

Optimal Control of Trading Algorithms: A General Impulse Control Approach

We propose a general framework for intraday trading based on the control of trading algorithms. Given a set of generic parameterized algorithms (which have to be specified by the controller ex-ante), our aim is to optimize the dates $(\tau_i)_i$ at which they are launched, the length $(\delta_i)_i$ of the trading period, and the value of the parameters $({\cal E}_i)_i$ kept during the time interval $[\tau_i,\tau_i + \delta_i)$. This provides the financial agent a decision tool for selecting which algorithm (and for which set of parameters and time length) should be used in the different phases of the trading period. From the mathematical point of view, this gives rise to a nonclassical impulse control problem where not only the regime ${\cal E}_i$ but also the period $[\tau_i,\tau_i+ \delta_i)$ have to be determined by the controller at the impulse time $\tau_i$. We adapt the weak dynamic programming principle of Bouchard and Touzi [SIAM J. Control Optim., 49 (2011), pp. 948–962] to our context to provide a characterization of the associated value function as a discontinuous viscosity solution of a system of partial differential equations with appropriate boundary conditions, for which we prove a comparison principle. We also propose a numerical scheme for the resolution of the above system and show that it is convergent. We finally provide a simple example of application to a problem of optimal stock trading with a nonlinear market impact function. This shows how parameters adapt to the market.

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.

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

We present a new fast-marching algorithm for an eikonal equation with a velocity-changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the fast-marching method in two respects. The first is that the new scheme can deal with a time-dependent velocity, and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm, and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in $\mathbb{R}^2$.

Équations d'Hamilton–Jacobi liées aux jeux différentiels avec coût de type supremum

In this Note we present a study of the existence and uniqueness to discontinuous viscosity solutions of Hamilton–Jacobi PDE related with differential games with supremum cost. To cite this article: O.-S. Serea, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation with a velocity which is a non-local quantity depending on the whole shape of the dislocation line. We study the special case where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.

Optimal Control of Hybrid Systems and a System of Quasi-Variational Inequalities

Optimal control of a class of hybrid dynamical systems is studied using the method of dynamic programming. It is proved that the value function is a discontinuous viscosity solution of the corresponding Hamilton--Jacobi--Bellman (HJB) equation which appears as a system of quasi-variational inequalities (SQVI) coupled by a nonlocal operator with a variable boundary condition. The comparison theorems are established for the sub- and super-solutions of the SQVI.

Existence and uniqueness for dislocation dynamics with nonnegative velocity

We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and nonlocal eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows us to prove that the energy associated to the dislocation dynamics is nonincreasing.

Stochastic targets with mixed diffusion processes and viscosity solutions

Let Z t, z ν be a R d+1 -valued mixed diffusion process controlled by ν with initial condition Z t, z ν ( t)= z. In this paper, we characterize the set of initial conditions such that Z t, z ν can be driven above a given stochastic target at time T by proving that the corresponding value function is a discontinuous viscosity solution of a variational partial differential equation. As applications of our main result, we study two examples: a problem of optimal insurance under self-protection and a problem of option hedging under jumping stochastic volatility where the underlying stock pays a random dividend at a fixed date.

Optimal stopping problems with discontinous reward: Regularity of the value function and viscosity solutions

We study optimal stopping problems for diffusion processes with discontinuous reward function. We give some results about the regularity of the value function and we show that, under suitable mild conditions on the underlying process, it has the same regularity of the reward function, namely, it is lower (respectively: upper) semicontinuous if the reward function is. The proofs for the two cases are quite different, and the upper semicontinuous case requires stronger conditions. Finally, we show that, in the case of lower semicontinuous reward, under suitable conditions the value function is a (discontinuous) viscosity solution of the associated variational inequalities.

A counter-example to the characterization of the discontinuous value function of control problems with reflection

We consider a finite horizon deterministic optimal control problem with reflection. The final cost is assumed to be merely a locally bounded function which leads to a discontinuous value function. We address the question of the characterization of the value function as the unique solution of an Hamilton–Jacobi equation with Neumann boundary conditions. We follow the discontinuous approach developed by Barles and Perthame for problems set in the whole space. We prove that the minimal and maximal discontinuous viscosity solutions of the associated Hamilton–Jacobi can be written in terms of value functions of control problems with reflection. Nethertheless, we construct a counter-example showing that the value function is not the unique solution of the equation. To cite this article: O. Ley, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 469–473.

Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions

In this paper, we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward SDEs and forward-backward SDEs. The controlled process $(X^\nu,Y^\nu)$ takes values in ${\mathbb R}^d \times {\mathbb R}$ and a given initial data for $X^{\nu}(0)$. Then the control problem is to find the minimal initial data for $Y^{\nu}$ so that it reaches a stochastic target at a specified terminal time T. The main application is from financial mathematics, in which the process $X^{\nu}$ is related to stock price, $Y^{\nu}$ is the wealth process, and $\nu$ is the portfolio. We introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation. The boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense. This provides a unique characterization of the value function which is the minimal initial data for $Y^{\nu}$.

A LEVEL SET APPROACH TO SEMICONTINUOUS VISCOSITY SOLUTIONS FOR CAUCHY PROBLEMS

A level set formulation is presented to characterize a maximal solution of the Cauchy problem for the Hamilton-Jacobi equation with semicontinuous initial data in an explicit way. No convexity assumptions on Hamiltonians are imposed. The solution proposed in the present paper is interpreted as the level set of an auxiliary problem and called an L-solution. It turns out that our L-solution is consistent with a classical discontinuous viscosity solution and a bitateral viscosity solution. Moreover, our L-solution is unique and enjoy the comparison principle. The condition that initial data is really attained is also discussed.

Formula for a solution ofu t +H(u,Du) =g

We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equationut +H(u,Du) =g in ℝn x ℝ+ withu(x, 0) =u0(x). The HamiltonianH(s,p) is assumed to be convex and positively homogeneous of degree one inp for eachs in ℝ. IfH is non increasing ins, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.

Hopf-Lax type formula for sub- and supersolutions

We study the continuous as well as discontinuous viscosity solutions of a certain Hamilton-Jacobi equation, $u_t + H(u, D u)=0$ in $\mathbb R^{\,n} \times \mathbb R_+$ with $u(x,0)=u_0(x)$. We obtain explicit formulas for continuous as well as for the sub- and supersolutions. In the latter case, furthermore, if $ H(u,p) > 0 $ for $|p| \neq 0 $ then the supersolution becomes a solution.

Discontinuous Solutions of a Hamilton--Jacobi Equation with Infinite Speed of Propagation

We consider the Cauchy problem \begin{eqnarray*} u\Rn\times (0,+\infty),$} u\Rn\times\{t=0\},$} \end{eqnarray*} where A(x) is a positive locally Lipschitz map from ${\Bbb R}n$ to the space of symmetric matrices. Since no growth condition on A is assumed, pathological phenomena can occur. We study the problem in the framework of discontinuous viscosity solutions and we get some comparison results and a representation formula for the minimal solution of the problem.

Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit

Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit

Discontinuous viscosity solutions to dirichlet problems for hamilton-jacob1 equations with

We introduce a new formulation of Dirichlet problem for a class of first order, nonlinear equations containing the minimum time problem, whose solution is expected to be discontinuous. We prove existence, uniqueness and representation formulas for the solution in the sense of viscosity solutions. Our method relies on a new way of prescribing the boundary condition, the use of recent ideas of Barron-Jensen [8] and Barles [5] , and the derivation of a backwards dynamic programming principle. We use the same ideas to prove uniqueness for the usual Dicchlet type formulation, following Ishii [13] and Bales-Perthame [6], under additional regularity conditions on the domain.