Some convergence results for Howard's algorithm

International audienceThis paper deals with convergence results of Howard's algorithm for the resolution of \min_{a\in \cA} (B^a x - b^a)=0 where $B^a$ is a matrix, $b^a$ is a vector (possibly of infinite dimension), and \cA is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices $B^a$. In the particular case of an obstacle problem of the form $\min(A x - b,\, x-g)=0$ where $A$ is an $N\times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than $N$ iterations, instead of the usual $2^N$ bound. Still in the case of obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm (M. Hintermüller et al., {\em SIAM J. Optim.}, Vol 13, 2002, pp. 865-888). We also propose an Howard-type algorithm for a "double-obstacle" problem of the form $\max(\min(Ax-b,x-g),x-h)=0$. We finally illustrate the algorithms on the discretization of nonlinear PDE's arising in the context of mathematical finance (American option, and Merton's portfolio problem), and for the double-obstacle problem

Numerical approximation for a superreplication problem under gamma constraints

International audienceWe study a superreplication problem of European options with gamma constraints, in mathematical finance. The initially unbounded control problem is set back to a problem involving a viscosity PDE solution with a set of bounded controls. Then a numerical approach is introduced, inconditionnally stable with respect to the mesh steps. A generalized finite difference scheme is used since basic finite differences cannot work in our case. Numerical tests illustrate the validity of our approach

Error estimates for a stochastic impulse control problem

We obtain error bounds for monotone approximation schemes of a stochastic impulse control problem. This is an extension of the theory for error estimates for the Hamilton-Jacobi-Bellman equation. For obtaining these bounds we build a sequence of stochastic impulse control problems, and a sequence of monotone approximation schemes. Extending methods of Barles and Jakobsen , we give error estimate for each problem of the sequence. Using these bounds we obtain the result. We obtain the same estimate on the rate of convergence as in the equation without impulsions

Error estimates for stochastic differential games: the adverse stopping case

We obtain error bounds for monotone approximation schemes of a particular Isaacs equation. This is an extension of the theory for estimating errors for the Hamilton-Jacobi-Bellman equation. For obtaining upper error bound, we consider the ``Krylov regularization'' of the Isaacs equation to build an approximate sub-solution of the scheme. To get lower error bound we extend the method of Barles and Jakobsen which consists in introducing a switching system whose solutions are local super-solutions of the Isaacs equation

Numerical analysis of stoshastic control problems

L'objet de la thèse est l'étude des approximations numériques de différentes équations HJB associées à des problèmes de contrôle optimal stochastique. Dans la première partie on a étendu la thèorie des estimations d'erreurs à un problème de jeux différentiels et au cas du problème avec impulsions. Ce dernier a fait l'objet d'une mise en oeuvre numérique. Dans toute cette partie l'ensemble de contrôles est borné. Ensuite, dans la deuxième partie de la thèse on a étudié des problèmes de contrôle stochastique provenant de la finance et dont l'ensemble des contrôles est non borné, en particulier des problèmes de sur-couverture.PARIS-BIUSJ-Thèses (751052125) / SudocPARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF

Error estimates for a stochastic impulse control problem

International audienceWe obtain error bounds for monotone approximation schemes of a stochastic impulse control problem. This is an extension of the theory for error estimates for the Hamilton-Jacobi-Bellman equation. We obtain almost the same estimate on the rate of convergence as in the equation without impulsions [2], [3]. © 2007 Springer

Some Convergence Results for Howard's Algorithm

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