Agnès Tourin

A viscosity solutions approach to shape-from-shading

The problem of recovering a Lambertian surface from a single two-dimensional image may be written as a first-order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions. A new approach based on Hamilton–Jacobi–Bellman equations and viscosity solutions theories enables one to study non-uniqueness phenomenon and thus to characterize the surface among the various solutions.A consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle, and numerical results are presented.

Shape-from-shading, viscosity solutions and edges

SummaryThis article deals with the so-called Shape-from-Shading problem which arises when recovering a shape from a single image. The general case of a distribution of light sources illuminating a Lambertian surface is considered. This involves original definitions of three types of edges, mainly the apparent contours, the grazing light edges and the shadow edges. The elevation of the shape is expressed in terms of viscosity solution of a first-order Hamilton-Jacobi equation with various boundary conditions on these edges. Various existence and uniqueness results are presented.

Viscosity solutions of nonlinear integro-differential equations

Abstract We investigate the questions of the existence and uniqueness of viscosity solutions to the Cauchy problem for integro-differential PDEs with nonlinear integral term. The existence of a solution is established by considering semicontinuous subsolutions and supersolutions and applying Perron’s method. Uniqueness is proved for both bounded and unbounded solutions. These results are then applied to a problem arising in Finance, namely the stochastic differential utility model under mixed Poisson-Brownian information.

Optimal stochastic control, stochastic target problems, and backward SDE

Preface.- 1. Conditional Expectation and Linear Parabolic PDEs.- 2. Stochastic Control and Dynamic Programming.- 3. Optimal Stopping and Dynamic Programming.- 4. Solving Control Problems by Verification.- 5. Introduction to Viscosity Solutions.- 6. Dynamic Programming Equation in the Viscosity Sense.- 7. Stochastic Target Problems.- 8. Second Order Stochastic Target Problems.- 9. Backward SDEs and Stochastic Control.- 10. Quadratic Backward SDEs.- 11. Probabilistic Numerical Methods for Nonlinear PDEs.- 12. Introduction to Finite Differences Methods.- References.

Numerical schemes for investment models with singular transactions

This paper considers an infinite horizon investment-consumption model in which a single agent consumes and distributes his wealth between two assets, a bond and a stock. The problem of maximization of the total utility from consumption is treated, when state (amount allocated in assets) and control (consumption, rates of trading) constraints are present. The value function is characterized as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation which, actually, is a Variational Inequality with gradient constraints. Numerical schemes are then constructed in order to compute the value function and the location of the free boundaries of the so-called transaction regions. These schemes are a combination of implicit and explicit schemes; their convergence is obtained from the uniqueness of viscosity solutions to the HJB equation.

Dynamic Pairs Trading Using the Stochastic Control Approach

We propose a model for analyzing dynamic pairs trading strategies using the stochastic control approach. The model is explored in an optimal portfolio setting, where the portfolio consists of a bank account and two co-integrated stocks and the objective is to maximize for a fixed time horizon, the expected terminal utility of wealth. For the exponential utility function, we reduce the problem to a linear parabolic partial differential equation which can be solved in closed form. In particular, we exhibit the optimal positions in the two stocks.

Viscosity Solutions and Numerical Schemes for Investment / Consumption Models with Transaction Costs

Introduction In this article we examine a general investment and consumption decision problem for a single agent. The investor consumes at a nonnegative rate and he distributes his current wealth between two assets. One asset is a bond , i.e. a riskless security with instantaneous rate of return r . The other asset is a stock , whose price is driven by a Wiener process. When the investor makes a transaction, he pays transaction fees which are assumed to be proportional to the amount transacted. More specifically, let x t and y t be the investors holdings in the riskless and the risky security prior to a transaction at time t . If the investor increases (or decreases) the amount invested in the risky asset to y t + h t (or y t - h t ), the holding of the riskless asset decreases (increases) to x t - h t - λ h t (or y t + h t - µ h t ). The numbers λ and µ are assumed to be nonnegative and one of them must always be positive. The control objective is to maximize, in an infinite horizon, the expected discounted utility which comes only from consumption. Due to the presence of the transaction fees, this is a singular control problem. Our goals are to derive the Hamilton–Jacobi–Bellman (HJB) equation that the value function solves and to characterize the latter as its unique weak solution, to come up with numerical schemes which converge to the value function as well as the optimal investment and consumption rules and to perform actual numerical computations and compare the results to the ones obtained in closed form by Davis & Norman.

A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and application to Shape-from-Shading problems

SummaryThe reconstruction from a shaded image of a Lambertian and not self-shadowing surface illuminated by a single distant pointwise light source may be written as a first-order Hamilton-Jacobi equation.In this paper, we continue the investigation begun in E. Rouy and A. Tourin into the uniqueness of the solution of this equation; the approach is based on the viscosity solutions theory and the dynamic programming principle.More precisely, we concentrate here on the uniqueness of the viscosity solution of this equation in case the measured luminous intensity reflected by the surface is discontinuous along a smooth curve. We prove a general comparison result for a piecewise Lipschitz continuous Hamiltonian and illustrate it by numerical experiments.

Portfolio Selection with Transaction Costs

This paper considers an infinite horizon investment-consumption model in which a single agent consumes and distributes his wealth between two assets, a bond and a stock. The problem of maximization of the total utility from consumption is treated; State (amount allocated in assets) and control (consumption, rates of trading) constraints are present. It is shown that the value function is the unique viscosity solution of a variational inequality with gradient constraints. A monotone numerical scheme is then constructed in order to compute both the value function and the location of the free boundaries of the so-called transaction regions.

Optimal Soaring via Hamilton-Jacobi-Bellman Equations

Competition glider flying is a game of stochastic optimization, in which mathematics and quantitative strategies have historically played an important role. We address the problem of uncertain future atmospheric conditions by constructing a nonlinear Hamilton-Jacobi-Bellman equation for the optimal speed to fly, with a free boundary describing the climb/cruise decision. We consider two different forms of knowledge about future atmospheric conditions, the first in which the pilot has complete foreknowledge and the second in which the state of the atmosphere is a Markov process discovered by flying through it. We compute an accurate numerical solution by designing a robust monotone finite difference method. The results obtained are of direct applicability for glider flight.