An optimal stopping problem for a geometric Brownian motion with poissonian jumps

This paper examines an optimal stopping problem for a geometric Brownian motion with random jumps. It is assumed that jumps occur according to a time-homogeneous Poisson process and the proportions of these sizes are independent and identically distributed nonpositive random variables. The objective is to find an optimal stopping time of maximizing the expected discounted terminal reward which is defined as a nondecreasing power function of the stopped state. By applying the “smooth pasting technique” [1,2], we derive almost explicitly an optimal stopping rule of a threshold type and the optimal value function of the initial state. That is, we express the critical state of the optimal stopping region and the optimal value function by formulae which include only given problem parameters except an unknown to be uniquely determined by a nonlinear equation.     

Second-order invex functions in nonlinear programming

We introduce a notion of a second-order invex function. A Fréchet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Fréchet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.     

Dynamic Programming and Value-Function Approximation in Sequential Decision Problems: Error Analysis and Numerical Results

Value-function approximation is investigated for the solution via Dynamic Programming (DP) of continuous-state sequential N-stage decision problems, in which the reward to be maximized has an additive structure over a finite number of stages. Conditions that guarantee smoothness properties of the value function at each stage are derived. These properties are exploited to approximate such functions by means of certain nonlinear approximation schemes, which include splines of suitable order and Gaussian radial-basis networks with variable centers and widths. The accuracies of suboptimal solutions obtained by combining DP with these approximation tools are estimated. The results provide insights into the successful performances appeared in the literature about the use of value-function approximators in DP. The theoretical analysis is applied to a problem of optimal consumption, with simulation results illustrating the use of the proposed solution methodology. Numerical comparisons with classical linear approximators are presented.     

A Bennett concentration inequality and its application to suprema of empirical processesUne inégalité de concentration de type Bennett et son application aux maxima de processus empiriques

We introduce new concentration inequalities for functions on product spaces. They allow to obtain a Bennett type deviation bound for suprema of empirical processes indexed by upper bounded functions. The result is an improvement on Rio's version [6] of Talagrand's inequality [7] for equidistributed variables. To cite this article: O. Bousquet, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 495–500.     

Interactive multiobjective complex search

An interactive procedure based on Box's complex search is used to solve the vector maximization problem. This method has the advantage that the decision maker's underlying value function need not be explicitly specified. Also, the problem may have nonlinear objective functions and nonlinear constraints. Several example problems are presented.     

Possibilities of solution in stochastic decision models with recursive reward functions

A stochastic decision model with general, non-necessarily additive reward function is considered and essential properties of such reward functions are formulated which only allow a successive proceeding in the sense of dynamic programming. Conditions for recursive—additive reward functions are given which ensure the existence of optimal strategies and the usability of value iteration to find an optimal policy and the optimal total reward.     

Sufficient conditions for the solubility of inverse eigenvalue problems

We define an inverse eigenvalue problem, which contains as special cases the classical additive and multiplicative inverse eigenvalue problems. Using some results on the distance of eigenvalues from matrix diagonal elements and Brouwer's fixed-point theorem, we give sufficient conditions for the solubility of the problem.     

Exponential asymptotics and adiabatic invariance of a simple oscillator

An alternative proof is provided for Littlewood's asymptotic expression arising from Lorentz's problem (1911) on the adiabatic invariance of a simple pendulum. Our approach is based on a standard WKB approximation. Our proof is simpler than those of both Littlewood (1963) and Wasow (1973). If the coefficient function in their differential equation is analytic, then Littlewood's asymptotic expression can even be replaced by an exponentially small term. To cite this article: C.H. Ou, R. Wong, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Inverse multi-objective combinatorial optimization

Inverse multi-objective combinatorial optimization consists of finding a minimal adjustment of the objective functions coefficients such that a given set of feasible solutions becomes efficient. An algorithm is proposed for rendering a given feasible solution into an efficient one. This is a simplified version of the inverse problem when the cardinality of the set is equal to one. The adjustment is measured by the Chebyshev distance. It is shown how to build an optimal adjustment in linear time based on this distance, and why it is right to perform a binary search for determining the optimal distance. These results led us to develop an approach based on the resolution of mixed-integer linear programs. A second approach based on a branch-and-bound is proposed to handle any distance function that can be linearized. Finally, the initial inverse problem is solved by a cutting plane algorithm.     

Some results on the uniqueness of generators of backward stochastic differential equations

It is proved that the generator g of a backward stochastic differential equation (BSDE) can be uniquely determined by the initial values of the corresponding BSDEs with all terminal conditions. The main results also confirm and extend Peng's conjecture. To cite this article: L. Jiang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the convergence of a class of outer approximation algorithms for convex programs

This paper presents a new class of outer approximation methods for solving general convex programs. The methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem. Moreover, the methods employ quadratic objective functions in the subproblems by adding a simple quadratic term to the objective function of the original problem, while other outer approximation methods usually use the original objective function itself throughout the iterations. By this modification, convergence of the methods can be proved under mild conditions. Furthermore, it is shown that generalized versions of the cut construction schemes in Kelley-Cheney-Goldstein's cutting plane method and Veinott's supporting hyperplane method can be incorporated with the present methods and a cut generated at each iteration need not be retained in the succeeding iterations.     

Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions

In this paper, we present some inverse function theorems and implicit function theorems for set-valued mappings between Fréchet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence of solutions of differential equations in Fréchet spaces with non-smooth data is given.     

Marches aléatoires et théorie du potentiel dans les domaines lipschitziens

We give a technical estimate on the gradients of the Green's functions in Lipschitz domains. The main application is a sharp Central Limit Theorem for random walks in these domains. To cite this article: N.Th. Varopoulos, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The dynamic programming method in the generalized traveling salesman problem

A procedure of the dynamic programming (DP) for the discrete-continuous problem of a route optimization is considered. It is possible to consider this procedure as a dynamic method of optimization of the towns choice in the well-known traveling salesman problem. In the considered version of DP, elements of a dynamic optimization are used. Two variants of the function of the aggregations of losses are investigated:

• 1.(1) the additive functions;
• 2.(2) the function characterizing the aggregation of losses in the bottle-neck problem.

     

Objective comparisons of the optimal portfolios corresponding to different utility functions

This paper considers the effects of some frequently used utility functions in portfolio selection by comparing the optimal investment outcomes corresponding to these utility functions. Assets are assumed to form a complete market of the Black–Scholes type. Under consideration are four frequently used utility functions: the power, logarithm, exponential and quadratic utility functions. To make objective comparisons, the optimal terminal wealths are derived by integration representation. The optimal strategies which yield optimal values are obtained by the integration representation of a Brownian martingale. The explicit strategy for the quadratic utility function is new. The strategies for other utility functions such as the power and the logarithm utility functions obtained this way coincide with known results obtained from Merton’s dynamic programming approach.     

Central Limit Theorem for Nonlinear Semi-Markov Reward Processes

Abstract

In this work, we obtain a central limit theorem for reward processes defined on a finite state space semi-Markov process, when reward functions assumed to have general forms and are not of constant rates. Martingale theory is the main tool which have been used for establishing the convergence of scaled and shifted reward process to a zero mean Brownian motion. The striking point in this article is considering general forms for the reward functions which are realistic in applications. The conditions needed for these results are existence of variances for sojourn times in each state and second order integrability of reward functions with respect to sojourn times distributions.     

On the Rédei zeta function

Let L be a locally finite lattice. An order function ν on L is a function defined on pairs of elements x, y (with x ≤ y) in L such that ν(x, y) = ν(x, z) ν(z, y). The Rédei zeta function of L is given by ?(s; L) = Σ_x∈Lμ(Ô, x) ν(Ô, x)^?s. It generalizes the following functions: the chromatic polynomial of a graph, the characteristic polynomial of a lattice, the inverse of the Dedekind zeta function of a number field, the inverse of the Weil zeta function for a variety over a finite field, Philip Hall's φ-function for a group and Rédei's zeta function for an abelian group. Moreover, the paradigmatic problem in all these areas can be stated in terms of the location of the zeroes of the Rédei zeta function.     

Bayesian Inversion of Piecewise Affine Operators in a Gaussian Framework

Piecewise affine inverse problems form a general class of nonlinear inverse problems. In particular inverse problems obeying certain variational structures, such as Fermat's principle in travel time tomography, are of this type. In a piecewise affine inverse problem a parameter is to be reconstructed when its mapping through a piecewise affine operator is observed, possibly with errors. A piecewise affine operator is defined by partitioning the parameter space and assigning a specific affine operator to each part. A Bayesian approach with a Gaussian random field prior on the parameter space is used. Both problems with a discrete finite partition and a continuous partition of the parameter space are considered.

The main result is that the posterior distribution is decomposed into a mixture of truncated Gaussian distributions, and the expression for the mixing distribution is partially analytically tractable. The general framework has, to the authors' knowledge, not previously been published, although the result for the finite partition is generally known.

Inverse problems are currently of large interest in many fields. The Bayesian approach is popular and most often highly computer intensive. The posterior distribution is frequently concentrated close to high-dimensional nonlinear spaces, resulting in slow mixing for generic sampling algorithms. Inverse problems are, however, often highly structured. In order to develop efficient sampling algorithms for a problem at hand, the problem structure must be exploited.

The decomposition of the posterior distribution that is derived in the current work can be used to develop specialized sampling algorithms. The article contains examples of such sampling algorithms. The proposed algorithms are applicable also for problems with exact observations. This is a case for which generic sampling algorithms tend to fail.     

Uniform limit laws of the logarithm for nonparametric estimators of the regression function in presence of censored data

In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions we consider. Examples of confidence bands obtained from simulated data are displayed.      

Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds,or better

This paper will do the following: (1) Establish a (better than) Thue-Siegel-Roth-Schmidt theorem bounding the approximation of solutions of linear differential equations over valued differential fields; (2) establish an effective better than Thue-Siegel-Roth-Schmidt theorem bounding the approximation of irrational algebraic functions (of one variable over a constant field of characteristic zero) by rational functions; (3) extend Nevanlinna's Three Small Function Theorem to an n small function theorem (for each positve integer n), by removing Chuang's dependence of the bound upon the relative “number” of poles and zeros of an auxiliary function; (4) extend this n Small Function Theorem to the case in which the n small functions are algebroid (a case which has applications in functional equations); (5) solidly connect Thue-Siegel-Roth-Schmidt approximation theory for functions with many of the Nevanlinna theories. The method of proof is (ultimately) based upon using a Thue-Siegel-Roth-Schmidt type auxiliary polynomial to construct an auxiliary differential polynomial.