On near-optimal necessary and sufficient conditions for forward-backward stochastic systems with jumps,with applications to finance

We establish necessary and sufficient conditions of near-optimality for nonlinear systems governed by forward-backward stochastic differential equations with controlled jump processes (FBSDEJs in short). The set of controls under consideration is necessarily convex. The proof of our result is based on Ekeland’s variational principle and continuity in some sense of the state and adjoint processes with respect to the control variable. We prove that under an additional hypothesis, the near-maximum condition on the Hamiltonian function is a sufficient condition for near-optimality. At the end, as an application to finance, mean-variance portfolio selection mixed with a recursive utility optimization problem is given. Mokhtar Hafay     

Sufficient and necessary conditions for stochastic near-optimal controls: A stochastic chemostat model with non-zero cost inhibiting

Near-optimal controls are as important as optimal controls for both theory and applications. Meanwhile, using inhibitor to control harmful microorganisms and ensure maximum growth of beneficial microorganisms (target microorganisms) is a very interesting topic in the chemostat. Thus, in this paper, we consider a stochastic chemostat model with non-zero cost inhibiting in finite time. The near-optimal control problem was constructed by minimizing the number of harmful microorganisms and minimizing the cost of inhibitor. We find that the Hamiltonian function is key to estimate objective function, and according to the adjoint equation, we obtain some error estimations of the near-optimality. Finally, we establish sufficient and necessary conditions for stochastic near-optimal controls of this model and numerical simulations and some conclusions are given.     

A sufficient condition for near-optimal stochastic controls and its application to manufacturing systems

This paper presents an asymptotic analysis of control models governed by stochastic ordinary differential equations. A sufficient condition of near-optimal controls is given based on Ekeland's principle. It is shown that, under some concavity assumptions, the-maximum condition in terms of the Hamiltonian implies the -optimality. By applying this result to a general manufacturing system, the common practice of hierarchical controls employed in order to reduce the overall complexity of the system is justified on a rigorous basis. A near-optimal control for the operational level is constructed from a near-optimal control at the corporate level, and an asymptotic error bound is obtained. A stochastic extension of the classical HMMS model is treated as a specific example. The approach of this paper is different from those in the literature, and it allows us to handle some previously unsolved problems with nonlinear state equations as well as nonseparable cost functions.This work was partly supported by NSERC Grant A4619, URIF, and the Manufacturing Research Corporation of Ontario.     

On necessary and sufficient conditions in vector optimization

It is shown that some general multiplier rules are necessary conditions for vector optimization in infinite-dimensional spaces. Under additional convexity assumptions, these conditions are sufficient. As an application, the Pontryagin maximum principle for cooperative differential games is examined.The authors are grateful to Professor W. Stadler and the referees of the previous edition of this paper for their valuable remarks and suggestions, which have been very helpful in the preparation of this paper.     

Some results on pointwise second-order necessary conditions for stochastic optimal controls

The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.     

On necessary and sufficient optimality conditions for multicriterial optimization problems

In this paper we derive first order necessary and sufficient optimality conditions for nonsmooth optimization problems with multiple criteria. These conditions are given for different optimality notions (i.e. weak, Pareto- and proper minimality) and for different types of derivatives of nonsmooth objective functions (locally Lipschitz continuous and quasidifferentiable) mappings. The conditions are given, if possible, in terms of a derivative and a subdifferential of those mappings.     

On necessary and sufficient conditions for the self-normalized central limit theorem

Let X₁, X₂, … be a sequence of independent random variables and S_n = Σ _i=1 ⁿ X_i and V _n ² = Σ _i=1 ⁿ X _i ² . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n=V_n converges to a standard normal distribution if and only if max_1?i?n|X_i|/V_n→0 in probability and the mean of X₁ is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max_1?i?n|X_i|/V_n→0 in probability, then these sufficient conditions are necessary.     

零对称BZ-代数的充要条件

重新证明文[10]中几个重要结论并修正文[10]中的定理1(11)和定理2．在此基础上，利用这些重新证明过的结论及修正过的定理可以按照文[10]中引理3，定理4，定理6，定理7，定理10的证明过程原样证明文[10]中的相应结果．因而在文[10]中，除性质11是结合BZ一代数的等价性质(见文[15])，定理1(11)及定理2需要进行修正外，其余结论及证明过程均成立．     

Optimality necessary conditions in singular stochastic control problems with nonsmooth data

The present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control problems with smooth coefficients. We, then apply Ekeland's variational principle for this approximating sequence of control problems, in order to establish necessary conditions satisfied by a sequence of near optimal controls. Finally, we prove the convergence of the scheme, using Krylov's inequality in the nondegenerate case and the Bouleau-Hirsch flow property in the degenerate one. The adjoint process obtained is given by means of distributional derivatives of the coefficients.     

On sufficient and necessary conditions for the Jacobi matrix inverse eigenvalue problem

Summary. In this paper, we study the inverse eigenvalue problem of a specially structured Jacobi matrix, which arises from the discretization of the differential equation governing the axial of a rod with varying cross section (Ram and Elhay 1998 Commum. Numer. Methods Engng. 14 597-608). We give a sufficient and some necessary conditions for such inverse eigenvalue problem to have solutions. Based on these results, a simple method for the reconstruction of a Jacobi matrix from eigenvalues is developed. Numerical examples are given to demonstrate our results.Research supported in part by National Natural Science Foundation of ChinaResearch supported in part by RGC Grant Nos. 7130/02P and 7046/03P, and HKU CRCG Grant Nos 10203501, and 10204437     

Some necessary and sufficient conditions for Hypercyclicity Criterion

We give necessary and sufficient conditions for an operator on a separable Hilbert space to satisfy the hypercyclicity criterion. This paper is a part of the second author’s Doctoral thesis, written at Shiraz University under the direction of the first author.     

On necessary and sufficient conditions for numerical verification of double turning points

Summary. This paper describes numerical verification of a double turning point of a nonlinear system using an extended system. To verify the existence of a double turning point, we need to prove that one of the solutions of the extended system corresponds to the double turning point. For that, we propose an extended system with an additional condition. As an example, for a finite dimensional problem, we verify the existence and local uniqueness of a double turning point numerically using the extended system and a verification method based on the Banach fixed point theorem.Mathematics Subject Classification (2000): 65J15, 65G20, 65P30     

Necessary optimality conditions for singular controls

A definition of singular controls with respect to components is given that includes, in particular, the conventional definition. On the basis of this definition, new necessary optimiality conditions for singular controls with respect to components are derived for the processes governed by systems of ordinary differential equations.     

A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions

In this paper, we concert with the existence of positive solution for the following nonlinear singular differential system with four-point boundary conditions

     

Deterministic near-optimal control,part 1: Necessary and sufficient conditions for near-optimality

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal control, for systems governed by deterministic ordinary differential equations. Necessary and sufficient conditions for near-optima control are studied. It is shown that any near-optimal control nearly maximizes the Hamiltonian in some integral sense, and vice versa, if some additional concavity conditions are imposed. Error estimates for both the near-optimality of the controls and the near-maximality of the Hamiltonian are obtained. A number of examples are presented to illustrate these results.This work was supported by the RGC Earmarked Grant CUHK 249/94E. Helpful comments from L. D. Berkovitz are gratefully acknowledged.     

General necessary and sufficient conditions for constrained optimum problems

Dedicated to Professor Heinz König on the occasion of his 65th birthday     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Necessary conditions for optimality of singular controls

The present paper is concerned with the study of controls which are singular in the sense of the maximum principle. We obtain necessary conditions for optimality of singular controls in systems governed by ordinary differential equations. A useful feature of the method considered here is that it can be applied to optimal control problems with distributed parameters.this research was supported in part by the National Science Foundation under Grant No. NSF-MCS-80-02337 at the University of Michigan.The author wishes to express his deep gratitude to Professor L. Cesari for his valuable guidance and constant encouragement during the preparation of this paper.