**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1735

# Search results for: extended Hamilton Jacobi Bellman equations

##### 1735 A Study of Hamilton-Jacobi-Bellman Equation Systems Arising in Differential Game Models of Changing Society

**Authors:**
Weihua Ruan,
Kuan-Chou Chen

**Abstract:**

**Keywords:**
Differential games,
Hamilton-Jacobi-Bellman
equations,
infinite horizon,
political-economy models.

##### 1734 A New Approach to the Approximate Solutions of Hamilton-Jacobi Equations

**Authors:**
Joe Imae,
Kenjiro Shinagawa,
Tomoaki Kobayashi,
Guisheng Zhai

**Abstract:**

We propose a new approach on how to obtain the approximate solutions of Hamilton-Jacobi (HJ) equations. The process of the approximation consists of two steps. The first step is to transform the HJ equations into the virtual time based HJ equations (VT-HJ) by introducing a new idea of ‘virtual-time’. The second step is to construct the approximate solutions of the HJ equations through a computationally iterative procedure based on the VT-HJ equations. It should be noted that the approximate feedback solutions evolve by themselves as the virtual-time goes by. Finally, we demonstrate the effectiveness of our approximation approach by means of simulations with linear and nonlinear control problems.

**Keywords:**
Nonlinear Control,
Optimal Control,
Hamilton-Jacobi Equation,
Virtual-Time

##### 1733 Effect of Supplementary Premium on the Optimal Portfolio Policy in a Defined Contribution Pension Scheme with Refund of Premium Clauses

**Authors:**
Edikan E. Akpanibah Obinichi C. Mandah Imoleayo S. Asiwaju

**Abstract:**

In this paper, we studied the effect of supplementary premium on the optimal portfolio policy in a defined contribution (DC) pension scheme with refund of premium clauses. This refund clause allows death members’ next of kin to withdraw their relative’s accumulated wealth during the accumulation period. The supplementary premium is to help sustain the scheme and is assumed to be stochastic. We considered cases when the remaining wealth is equally distributed and when it is not equally distributed among the remaining members. Next, we considered investments in cash and equity to help increase the remaining accumulated funds to meet up with the retirement needs of the remaining members and composed the problem as a continuous time mean-variance stochastic optimal control problem using the actuarial symbol and established an optimization problem from the extended Hamilton Jacobi Bellman equations. The optimal portfolio policy, the corresponding optimal fund size for the two assets and also the efficient frontier of the pension members for the two cases was obtained. Furthermore, the numerical simulations of the optimal portfolio policies with time were presented and the effect of the supplementary premium on the optimal portfolio policy was discussed and observed that the supplementary premium decreases the optimal portfolio policy of the risky asset (equity). Secondly we observed a disparity between the optimal policies for the two cases.

**Keywords:**
Defined contribution pension scheme,
extended Hamilton Jacobi Bellman equations,
optimal portfolio policies,
refund of premium clauses,
supplementary premium.

##### 1732 Application of Legendre Transformation to Portfolio Optimization

**Authors:**
Peter Benneth,
Tsaroh N. Theophilus,
Prince Benjamin

**Abstract:**

This research work aims at studying the application of Legendre Transformation Method (LTM) to Hamilton Jacobi Bellman (HJB) equation which is an example of optimal control problem. We discuss the steps involved in modelling the HJB equation as it relates to mathematical finance by applying the Ito’s lemma and maximum principle theorem. By applying the LTM and dual theory, the resultant HJB equation is transformed to a linear Partial Differential Equation (PDE). Also, the Optimal Investment Strategy (OIS) and the optimal value function were obtained under the exponential utility function. Furthermore, some numerical results were also presented with observations that the OIS under exponential utility is directly proportional to the appreciation rate of the risky asset and inversely proportional to the instantaneous volatility, predetermined interest rate, risk averse coefficient. Finally, it was observed that the optimal fund size is an increasing function of the risk free interest rate. This result is consistent with some existing results.

**Keywords:**
Legendre transformation method,
Optimal investment strategy,
Ito’s lemma,
Hamilton Jacobi Bellman equation,
Geometric Brownian motion,
financial market.

##### 1731 Mean-Variance Optimization of Portfolios with Return of Premium Clauses in a DC Pension Plan with Multiple Contributors under Constant Elasticity of Variance Model

**Authors:**
Bright O. Osu,
Edikan E. Akpanibah,
Chidinma Olunkwa

**Abstract:**

In this paper, mean-variance optimization of portfolios with the return of premium clauses in a defined contribution (DC) pension plan with multiple contributors under constant elasticity of variance (CEV) model is studied. The return clauses which permit death members to claim their accumulated wealth are considered, the remaining wealth is not equally distributed by the remaining members as in literature. We assume that before investment, the surplus which includes funds of members who died after retirement adds to the total wealth. Next, we consider investments in a risk-free asset and a risky asset to meet up the expected returns of the remaining members and obtain an optimized problem with the help of extended Hamilton Jacobi Bellman equation. We obtained the optimal investment strategies for the two assets and the efficient frontier of the members by using a stochastic optimal control technique. Furthermore, we studied the effect of the various parameters of the optimal investment strategies and the effect of the risk-averse level on the efficient frontier. We observed that the optimal investment strategy is the same as in literature, secondly, we observed that the surplus decreases the proportion of the wealth invested in the risky asset.

**Keywords:**
DC pension fund,
Hamilton Jacobi Bellman equation,
optimal investment strategies,
stochastic optimal control technique,
return of premiums clauses,
mean-variance utility.

##### 1730 Optimal Portfolio Selection in a DC Pension with Multiple Contributors and the Impact of Stochastic Additional Voluntary Contribution on the Optimal Investment Strategy

**Authors:**
Edikan E. Akpanibah,
Okwigbedi Oghen’Oro

**Abstract:**

In this paper, we studied the optimal portfolio selection in a defined contribution (DC) pension scheme with multiple contributors under constant elasticity of variance (CEV) model and the impact of stochastic additional voluntary contribution on the investment strategies. We assume that the voluntary contributions are stochastic and also consider investments in a risk free asset and a risky asset to increase the expected returns of the contributing members. We derived a stochastic differential equation which consists of the members’ monthly contributions and the invested fund and obtained an optimized problem with the help of Hamilton Jacobi Bellman equation. Furthermore, we find an explicit solution for the optimal investment strategy with stochastic voluntary contribution using power transformation and change of variables method and the corresponding optimal fund size was obtained. We discussed the impact of the voluntary contribution on the optimal investment strategy with numerical simulations and observed that the voluntary contribution reduces the optimal investment strategy of the risky asset.

**Keywords:**
DC pension fund,
Hamilton-Jacobi-Bellman,
optimal investment strategies,
power transformation method,
stochastic,
voluntary contribution.

##### 1729 An Insurer’s Investment Model with Reinsurance Strategy under the Modified Constant Elasticity of Variance Process

**Authors:**
K. N. C. Njoku,
Chinwendu Best Eleje,
Christian Chukwuemeka Nwandu

**Abstract:**

One of the problems facing most insurance companies is how best the burden of paying claims to its policy holders can be managed whenever need arises. Hence there is need for the insurer to buy a reinsurance contract in order to reduce risk which will enable the insurer to share the financial burden with the reinsurer. In this paper, the insurer’s and reinsurer’s strategy is investigated under the modified constant elasticity of variance (M-CEV) process and proportional administrative charges. The insurer considered investment in one risky asset and one risk free asset where the risky asset is modeled based on the M-CEV process which is an extension of constant elasticity of variance (CEV) process. Next, a nonlinear partial differential equation in the form of Hamilton Jacobi Bellman equation is obtained by dynamic programming approach. Using power transformation technique and variable change, the explicit solutions of the optimal investment strategy and optimal reinsurance strategy are obtained. Finally, some numerical simulations of some sensitive parameters were obtained and discussed in details where we observed that the modification factor only affects the optimal investment strategy and not the reinsurance strategy for an insurer with exponential utility function.

**Keywords:**
Reinsurance strategy,
Hamilton Jacobi Bellman equation,
power transformation,
M-CEV process,
exponential utility.

##### 1728 Jacobi-Based Methods in Solving Fuzzy Linear Systems

**Authors:**
Lazim Abdullah,
Nurhakimah Ab. Rahman

**Abstract:**

Linear systems are widely used in many fields of science and engineering. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. Therefore it is important to perform numerical algorithms or procedures that would treat general fuzzy linear systems and solve them using iterative methods. This paper aims are to solve fuzzy linear systems using four types of Jacobi based iterative methods. Four iterative methods based on Jacobi are used for solving a general n × n fuzzy system of linear equations of the form Ax = b , where A is a crisp matrix and b an arbitrary fuzzy vector. The Jacobi, Jacobi Over-Relaxation, Refinement of Jacobi and Refinement of Jacobi Over-Relaxation methods was tested to a five by five fuzzy linear system. It is found that all the tested methods were iterated differently. Due to the effect of extrapolation parameters and the refinement, the Refinement of Jacobi Over-Relaxation method was outperformed the other three methods.

**Keywords:**
Fuzzy linear systems,
Jacobi,
Jacobi Over-
Relaxation,
Refinement of Jacobi,
Refinement of Jacobi Over-
Relaxation.

##### 1727 Complexity Reduction Approach with Jacobi Iterative Method for Solving Composite Trapezoidal Algebraic Equations

**Authors:**
Mohana Sundaram Muthuvalu,
Jumat Sulaiman

**Abstract:**

In this paper, application of the complexity reduction approach based on half- and quarter-sweep iteration concepts with Jacobi iterative method for solving composite trapezoidal (CT) algebraic equations is discussed. The performances of the methods for CT algebraic equations are comparatively studied by their application in solving linear Fredholm integral equations of the second kind. Furthermore, computational complexity analysis and numerical results for three test problems are also included in order to verify performance of the methods.

**Keywords:**
Complexity reduction approach,
Composite trapezoidal
scheme,
Jacobi method,
Linear Fredholm integral equations

##### 1726 On the Algorithmic Iterative Solutions of Conjugate Gradient, Gauss-Seidel and Jacobi Methods for Solving Systems of Linear Equations

**Authors:**
H. D. Ibrahim,
H. C. Chinwenyi,
H. N. Ude

**Abstract:**

In this paper, efforts were made to examine and compare the algorithmic iterative solutions of conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax = b, where A is a real n x n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3 x 3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi and Conjugate Gradient methods) respectively. From the results obtained, we discovered that the Conjugate Gradient method converges faster to exact solutions in fewer iterative steps than the two other methods which took much iteration, much time and kept tending to the exact solutions.

**Keywords:**
conjugate gradient,
linear equations,
symmetric and positive definite matrix,
Gauss-Seidel,
Jacobi,
algorithm

##### 1725 Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations

**Authors:**
Fuziyah Ishak,
Siti Norazura Ahmad

**Abstract:**

Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

**Keywords:**
Accuracy,
extended trapezoidal method,
numerical solution,
Volterra integro-differential equations.

##### 1724 Preconditioned Jacobi Method for Fuzzy Linear Systems

**Authors:**
Lina Yan,
Shiheng Wang,
Ke Wang

**Abstract:**

A preconditioned Jacobi (PJ) method is provided for solving fuzzy linear systems whose coefficient matrices are crisp Mmatrices and the right-hand side columns are arbitrary fuzzy number vectors. The iterative algorithm is given for the preconditioned Jacobi method. The convergence is analyzed with convergence theorems. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.

**Keywords:**
preconditioning,
M-matrix,
Jacobi method,
fuzzy linear
system (FLS).

##### 1723 Advanced Gronwall-Bellman-Type Integral Inequalities and Their Applications

**Authors:**
Zixin Liu,
Shu Lü,
Shouming Zhong,
Mao Ye

**Abstract:**

**Keywords:**
Gronwall-Bellman-Type integral inequalities,
integrodifferential equation,
p-exponentially stable,
mixed delays.

##### 1722 Modelling an Investment Portfolio with Mandatory and Voluntary Contributions under M-CEV Model

**Authors:**
Amadi Ugwulo Chinyere,
Lewis D. Gbarayorks,
Emem N. H. Inamete

**Abstract:**

In this paper, the mandatory contribution, additional voluntary contribution (AVC) and administrative charges are merged together to determine the optimal investment strategy (OIS) for a pension plan member (PPM) in a defined contribution (DC) pension scheme under the modified constant elasticity of variance (M-CEV) model. We assume that the voluntary contribution is a stochastic process and a portfolio consisting of one risk free asset and one risky asset modeled by the M-CEV model is considered. Also, a stochastic differential equation consisting of PPM’s monthly contributions, voluntary contributions and administrative charges is obtained. More so, an optimization problem in the form of Hamilton Jacobi Bellman equation which is a nonlinear partial differential equation is obtained. Using power transformation and change of variables method, an explicit solution of the OIS and the value function are obtained under constant absolute risk averse (CARA). Furthermore, numerical simulations on the impact of some sensitive parameters on OIS were discussed extensively. Finally, our result generalizes some existing result in the literature.

**Keywords:**
DC pension fund,
modified constant elasticity of variance,
optimal investment strategies,
voluntary contribution,
administrative charges.

##### 1721 The Convergence Results between Backward USSOR and Jacobi Iterative Matrices

**Authors:**
Zuan-De Wang,
Hou-biao Li,
Zhong-xi Gao

**Abstract:**

In this paper, the backward Ussor iterative matrix is proposed. The relationship of convergence between the backward Ussor iterative matrix and Jacobi iterative matrix is obtained, which makes the results in the corresponding references be improved and refined.Moreover,numerical examples also illustrate the effectiveness of these conclusions.

**Keywords:**
Backward USSOR iterative matrix,
Jacobi iterative matrix,
convergence,
spectral radius

##### 1720 The Relationship of Eigenvalues between Backward MPSD and Jacobi Iterative Matrices

**Authors:**
Zhuan-de Wang,
Hou-biao Li,
Zhong-xi Gao

**Abstract:**

In this paper, the backward MPSD (Modified Preconditioned Simultaneous Displacement) iterative matrix is firstly proposed. The relationship of eigenvalues between the backward MPSD iterative matrix and backward Jacobi iterative matrix for block p-cyclic case is obtained, which improves and refines the results in the corresponding references.

**Keywords:**
Backward MPSD iterative matrix,
Jacobi iterative matrix,
eigenvalue,
p-cyclic matrix.

##### 1719 Extended Arithmetic Precision in Meshfree Calculations

**Authors:**
Edward J. Kansa,
Pavel Holoborodko

**Abstract:**

Continuously differentiable radial basis functions (RBFs) are meshfree, converge faster as the dimensionality increases, and is theoretically spectrally convergent. When implemented on current single and double precision computers, such RBFs can suffer from ill-conditioning because the systems of equations needed to be solved to find the expansion coefficients are full. However, the Advanpix extended precision software package allows computer mathematics to resemble asymptotically ideal Platonic mathematics. Additionally, full systems with extended precision execute faster graphical processors units and field-programmable gate arrays because no branching is needed. Sparse equation systems are fast for iterative solvers in a very limited number of cases.

**Keywords:**
Meshless spectrally convergent,
partial differential equations,
extended arithmetic precision,
no branching.

##### 1718 Explicit Solution of an Investment Plan for a DC Pension Scheme with Voluntary Contributions and Return Clause under Logarithm Utility

**Authors:**
Promise A. Azor,
Avievie Igodo,
Esabai M. Ase

**Abstract:**

The paper merged the return of premium clause and voluntary contributions to investigate retirees’ investment plan in a defined contributory (DC) pension scheme with a portfolio comprising of a risk-free asset and a risky asset whose price process is described by geometric Brownian motion (GBM). The paper considers additional voluntary contributions paid by members, charge on balance by pension fund administrators and the mortality risk of members of the scheme during the accumulation period by introducing return of premium clause. To achieve this, the Weilbull mortality force function is used to establish the mortality rate of members during accumulation phase. Furthermore, an optimization problem from the Hamilton Jacobi Bellman (HJB) equation is obtained using dynamic programming approach. Also, the Legendre transformation method is used to transform the HJB equation which is a nonlinear partial differential equation to a linear partial differential equation and solves the resultant equation for the value function and the optimal distribution plan under logarithm utility function. Finally, numerical simulations of the impact of some important parameters on the optimal distribution plan were obtained and it was observed that the optimal distribution plan is inversely proportional to the initial fund size, predetermined interest rate, additional voluntary contributions, charge on balance and instantaneous volatility.

**Keywords:**
Legendre transform,
logarithm utility,
optimal distribution plan,
return clause of premium,
charge on balance,
Weibull mortality function.

##### 1717 Vibration of FGM Cylindrical Shells under Effect Clamped-simply Support Boundary Conditions using Hamilton's Principle

**Authors:**
M.R.Isvandzibaei,
E.Bidokh,
M.R.Alinaghizadeh,
A.Nasirian,
A.Moarrefzadeh

**Abstract:**

**Keywords:**
Vibration,
FGM,
Cylindrical shell,
Hamilton'sprinciple,
Ring support.

##### 1716 Linear Dynamic Stability Analysis of a Continuous Rotor-Disk-Blades System

**Authors:**
F. Rahimi Dehgolan,
S. E. Khadem,
S. Bab,
M. Najafee

**Abstract:**

**Keywords:**
Rotating shaft,
flexible blades,
centrifugal stiffening,
stability.

##### 1715 Strict Stability of Fuzzy Differential Equations with Impulse Effect

**Authors:**
Sanjay K.Srivastava,
Bhanu Gupta

**Abstract:**

In this paper some results on strict stability heve beeb extended for fuzzy differential equations with impulse effect using Lyapunov functions and Razumikhin technique.

**Keywords:**
Fuzzy differential equations,
Impulsive differential equations,
Strict stability,
Lyapunov function,
Razumikhin technique.

##### 1714 Vibration of Functionally Graded Cylindrical Shells Under Effect Clamped-Free Boundary Conditions Using Hamilton's Principle

**Authors:**
M.R. Isvandzibaei,
M.R. Alinaghizadeh,
A.H. Zaman

**Abstract:**

In the present work, study of the vibration of thin cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of boundary conditions on the natural frequencies of the FG cylindrical shell. The study is carried out using third order shear deformation shell theory. The analysis is carried out using Hamilton's principle. The governing equations of motion of FG cylindrical shells are derived based on shear deformation theory. Results are presented on the frequency characteristics, influence of constituent volume fractions and the effects of clamped-free boundary conditions

**Keywords:**
Vibration,
FGM,
cylindrical shell,
Hamilton's principle,
clamped supported.

##### 1713 New Application of EHTA for the Generalized(2+1)-Dimensional Nonlinear Evolution Equations

**Authors:**
Mohammad Taghi Darvishi,
Maliheh Najafi,
Mohammad Najafi

**Abstract:**

In this paper, the generalized (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (shortly CBS) equations are investigated. We employ the Hirota-s bilinear method to obtain the bilinear form of CBS equations. Then by the idea of extended homoclinic test approach (shortly EHTA), some exact soliton solutions including breather type solutions are presented.

**Keywords:**
EHTA,
(2+1)-dimensional CBS equations,
(2+1)-dimensional breaking solution equation,
Hirota's bilinear form.

##### 1712 Effects Edge end Free-free Boundary Conditions for Analysis Free Vibration of Functionally Graded Cylindrical Shell with Ring based on Third Order Shear Deformation Theory using Hamilton's Principle

**Authors:**
M.R.Isvandzibaei,
P.J.Awasare

**Abstract:**

**Keywords:**
Vibration,
FGM,
Cylindrical shell,
Hamilton'sprinciple,
Ring support.

##### 1711 N-Sun Decomposition of Complete Graphs and Complete Bipartite Graphs

**Authors:**
R. Anitha,
R. S. Lekshmi

**Abstract:**

**Keywords:**
Hamilton cycle,
n-sun decomposition,
perfectmatching,
spanning tree.

##### 1710 Numerical Study of Iterative Methods for the Solution of the Dirichlet-Neumann Map for Linear Elliptic PDEs on Regular Polygon Domains

**Authors:**
A. G. Sifalakis,
E. P. Papadopoulou,
Y. G. Saridakis

**Abstract:**

**Keywords:**
Elliptic PDEs,
Dirichlet to Neumann Map,
Global Relation,
Collocation,
Iterative Methods,
Jacobi,
Gauss-Seidel,
GMRES,
Bi-CGSTAB.

##### 1709 Free Vibration Analysis of Functionally Graded Beams

**Authors:**
Gholam Reza Koochaki

**Abstract:**

**Keywords:**
Functionally graded beam,
Free vibration,
Hamilton's principle.

##### 1708 Almost Periodic Solution for an Impulsive Neural Networks with Distributed Delays

**Authors:**
Lili Wang

**Abstract:**

By using the estimation of the Cauchy matrix of linear impulsive differential equations and Banach fixed point theorem as well as Gronwall-Bellman’s inequality, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solution for an impulsive neural networks with distributed delays. An example is presented to illustrate the feasibility and effectiveness of the results.

**Keywords:**
Almost periodic solution,
Exponential stability,
Neural networks,
Impulses.

##### 1707 Existence and Exponential Stability of Almost Periodic Solution for Cohen-Grossberg SICNNs with Impulses

**Abstract:**

In this paper, based on the estimation of the Cauchy matrix of linear impulsive differential equations, by using Banach fixed point theorem and Gronwall-Bellman-s inequality, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solution for Cohen-Grossberg shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays and impulses. An example is given to illustrate the main results.

**Keywords:**
Almost periodic solution,
exponential stability,
neural networks,
impulses.

##### 1706 Flexible Follower Response of a Translating Cam with Four Different Profiles for Rise-Dwell-Fall-Dwell motion

**Authors:**
Jer-Rong Chang

**Abstract:**

**Keywords:**
translating cam,
flexible follower,
rise-dwell-falldwell,
response