Edikan Akpanibah - Academia.edu (original) (raw)

Papers by Edikan Akpanibah

International Journal of Mathematical and Computational Science , 2024

This research work aims at studying the application of Legendre Transformation Method (LTM) to Ha... more 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.

INTERNATIONAL JOURNAL OF APPLIED SCIENCES AND MATHEMATICAL THEORY (IJASMT ), 2024

In this paper, the insurer and reinsurer's strategy and the reinsurer's surplus were studied in t... more In this paper, the insurer and reinsurer's strategy and the reinsurer's surplus were studied in the presence of some random environmental noise on the risky asset under logarithm utility function. A portfolio with one risky and risk free asset was considered such that the risky asset follows the geometric Brownian motion (GBM). It was also assume that the claim process of the insurer is a stochastic differential equation, and the reinsurer can buy proportional reinsurance policy as a backup for their investment. The maximum principle theory and Ito's lemma were used to derive our optimization problem. The Legendre transformation and dual theory with variable separation technique were used to solve the optimization problem under logarithm utility to obtain the optimal reinsurer strategy (ORS), optimal reinsurer policy (ORP) and the reinsurer's surplus. More so, some numerical analyses were presented to discuss the effectof somesensitive parameters on the ORS and ORP. The ORS was observed to be a decreasing function of the instantaneous volatility, risk free interest rate but an increasing function of the appreciation rate of the of the risky asset. Furthermore, the relationship between the surplus process and time, risky asset and environmental noise was also given.

Barekeng, May 25, 2024

Article History: The recent Central Bank of Nigeria (CBN) 2023 redesigned naira notes is of good ... more Article History: The recent Central Bank of Nigeria (CBN) 2023 redesigned naira notes is of good benefits to strengthen the economy of the country by checking counterfeiting and hoarding of large volume of banknotes by the public. Despite all the efforts made by the CBN for citizens to enjoy the benefits of this implementation, most rural farmers were faced with adverse effects of uncertainties in the production and marketing of their agricultural produce due to lack of redesigned new naira notes in circulation. The adverse effects of these uncertainties are modeled as Advanced Stochastic Time-Delay Differential Equation (ASTDDE). The modeled equation is solved using Extended Second Derivative Block Backward Differentiation Formulae Method (ESDBBDFM) without the use of interpolation techniques in the evaluations of the delay term and noise term. In comparing the numerical results of this method with other existing methods in literature, the newly developed mathematical expressions for the evaluations of the delay term and the noise term in solving ASDDEs with the discrete schemes of ESDBBDFM gives better results for step number = 4 than step numbers = 2 and 3 by producing Least Minimum Absolute Random Error (LMARE) in a Lower Computational Processing Unit Time (LCPUT) faster than other existing methods that applied interpolation techniques in evaluations of the delay term and the noise term.

Communication in Physical Sciences, Mar 29, 2021

The aim of this paper is to study the optimal investment plans of a member in a defined contribut... more The aim of this paper is to study the optimal investment plans of a member in a defined contribution (DC) pension scheme with proportional administrative fee and tax on invested funds under logarithm utility function and Ornstein-Uhlenbeck (O-U) model. This is done by considering a portfolio consisting of a risk free asset (bank security) and two risky assets (stocks) where the stock market prices are driven by the Ornstein-Uhlenbeck (O-U) process. An optimization problem known as the Hamilton Jacobi Bellman (HJB) equation is obtained by maximizing the expected utility of the member's terminal wealth. Since the HJB equation is a non linear partial differential equation (PDE) and could be complex to solve, we use the Legendre transformation method and dual theory to reduce it to a linear PDE. By method of variable change and separation of variable, closed form solutions of the optimal investment plans are obtained using logarithm utility function. More so, sensitivity analysis of some parameters are carried out theoretically on the optimal investment plans with observations that apart from the changes experienced in the stock market prices caused by the O-U process, the optimal investment plans for the risky assets are inversely proportional to contribution rate, tax rate imposed on the invested fund , proportional administration fee, investment time , but directly proportional to the appreciation rate of the risky assets.

Journal of the Nigerian Association of Mathematical Physics, 2019

Asian Journal of Probability and Statistics, 2022

One of the major challenges faced by most pension fund managers in the defined pension (DC) schem... more One of the major challenges faced by most pension fund managers in the defined pension (DC) scheme is how best member’s contributions can be invested to yield maximum returns. To achieve this, there is need to model and developed a robust investment plan which takes into consideration the volatility of the stock market price, tax on investment on risky assets and the mortality risk of its members. Based on this, the optimal portfolio distribution of a DC pension scheme with return of premium clause is studied where the mortality force function is characterized by the Weibull model and the investment in risky asset is subject to a certain proportion of tax. A portfolio with a risk-free asset and a risky asset modeled by the geometric Brownian motion such that the remaining accumulations are equally distributed between the remaining members is considered. Furthermore, the game theoretic approach is used to establish an optimization problem from the extended Hamilton Jacobi Bellman (HJ...

This paper investigates the optimal investment strategies for a defined contribution pension fund... more This paper investigates the optimal investment strategies for a defined contribution pension fund with return clauses of premiums with interest under the mean-variance criterion. Using the actuarial symbol, we formalize the problem as a continuous time mean-variance stochastic optimal control. The pension fund manager considers investments in risk and risk-free assets to increase the remaining accumulated funds to meet the retirement needs of the remaining members. Using the variational inequalities methods, we established an optimized problem from the extended Hamilton–Jacobi–Bellman Equations and solved the optimized problem to obtain the optimal investment strategies for both risk-free and risky assets and also the efficient frontier of the pension member. Furthermore, we evaluated analytically and numerically the effect of various parameters of the optimal investment strategies on it. We observed that the optimal investment strategy for the risky asset decreases with an increase...

We studied optimal investment strategies for a plan contributor in a defined pension scheme, with... more We studied optimal investment strategies for a plan contributor in a defined pension scheme, with stochastic salary and extra contributions, under the affine interest rate model. We considered two cases; where the extra contribution rates are stochastic and constant. We considered investment in three different assets namely risk free asset (cash), zero coupon bonds and the risky asset (stock). Using Legendre transformation method and dual theory, we obtained the optimal investment strategies the three investments using exponential utility function for the two cases. The result shows that the strategies for the respective investments used when there is no extra contribution can be used when the extra contribution rate is constant as in [1] but cannot be used when it is stochastic. Clearly this gives the member and the fund manager good insight on how to invest to maximize profit with minimal risk once this condition arises.

Journal of Nonlinear Sciences and Applications, 2019

In this paper, we study optimal asset allocation strategy for a defined contribution (DC) pension... more In this paper, we study optimal asset allocation strategy for a defined contribution (DC) pension fund with return of premium clause under Heston's volatility model in mean-variance utility frame work. In this model, members' next of kin are allowed to withdraw their family members' accumulated premium with predetermined interest. Also, investments in one risk free asset and one risky asset are considered to help increase the accumulated funds of the remaining members in order to meet their retirement needs. Using the actuarial symbol, we formulize the problem as a continuous time mean-variance stochastic optimal control problem. We establish an optimization problem from the extended Hamilton Jacobi Bellman equations using the game theoretic approach and solve the optimization problem to obtain the optimal allocation strategy for the two assets, the optimal fund size and also the efficient frontier of the pension members. We analyze numerically the effect of some parameters on the optimal allocation strategy and deduce that as the initial wealth, predetermined interest rate and risk averse level increases, the optimal allocation policy for the risky asset (equity) decreases. Furthermore, we give a theoretical comparison of our result with an existing result and observed that the optimal allocation policy whose return is with predetermined interest is higher compared to that without predetermined interest.

Journal of Mathematical and Computational Science, 2021

In this paper, we established and carried-out the computational solution of some first order dela... more In this paper, we established and carried-out the computational solution of some first order delay differential equations (DDEs) using hybrid extended backward differentiation formulae method in block forms without the application of interpolation techniques in determining the delay term. The discrete schemes were worked-out through the linear multistep collocation technique by matrix inversion approach from the continuous construction of each step number which clearly demonstrated the order and error constants, consistency, zero stability, convergence and region of absolute stability of this method after investigations. The results obtained after the implementation of this method validate that the lower step number integrated with hybrid extended future points performed better than the higher step numbers integrated with hybrid extended future points when compared with the exact solutions and other existing methods.

International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2021

This work studies the optimal control strategy for a pension plan with refund clause of contribut... more This work studies the optimal control strategy for a pension plan with refund clause of contributions with predetermined interest under constant elasticity of variance (CEV) in a defined contribution (DC) pension plan. A model which mandates fund managers to refund dead members' accumulations with predetermined interest to their next of kin during the accumulation phase is considered. Also considered herein are investments in a bank security and stock where the stock market price is driven by the CEV model and the remaining accumulations are equally distributed between the remaining members. Furthermore, the game theoretic approach is use in establishing an optimization problem from the extended Hamilton Jacobi Bellman (HJB) equation which is a non-linear partial differential equation (PDE). Using mean variance utility function and method of variable separation, explicit solutions of the optimal control strategy and the efficient frontier are obtained. Finally, Numerical simulations and theoretical analysis are used to study the effect of the elasticity parameter β and some other parameters on the optimal control strategy with observations that the elasticity parameter affects the investment strategy of the fund manager significantly. Also, we observed that the optimal control strategy employed by the fund manager is inversely proportional to the risk aversion coefficient, initial fund size, instantaneous volatility and predetermined interest rate but directly proportional to time.

Canadian Journal of pure and Applied Sciences, 2020

This paper merged together the study of optimal control laws for a pension plan with and without ... more This paper merged together the study of optimal control laws for a pension plan with and without return clause under Heston volatility model. An investment model comprising of members' monthly contributions, return accumulations with risk free interest to dead members' families for the case with return clause and investment in one risk free asset and two risky assets is presented. Since the mean variance utility function is time inconsistent, the game theoretic approach is used to establish an optimization problem from the extended Hamilton Jacobi Bellman (HJB) equation. Furthermore, the optimal control laws for the three assets and the efficient frontier are obtained using variable separation method by solving the extended HJB equations. Finally, Numerical simulations were presented to demonstrate the effects of some parameters on the optimal control laws with observations that the optimal control law for risk free asset decreases continuously with time while that of the risky assets increases continuously with time.

International Journal of basic science and technology, 2020

Considering the adverse effect of Corona virus on the financial market, there is need for banks t... more Considering the adverse effect of Corona virus on the financial market, there is need for banks to develop efficient portfolio strategy that is compact and takes into consideration the volatility of the stock market price. As a result of this, an efficient portfolio management for a commercial bank under constant elasticity of variance (CEV) model is studied using exponential utility function. A portfolio comprising of treasury security, marketable security and a loan is considered such that the last two assets are modelled by CEV model. Furthermore, the power transformation and change of variable technique are used to obtain explicit solutions of the optimal portfolio strategies, value function, bank’s total assets, deposits and capital with numerical simulations. Finally, based on the investment strategies employ by the bank, the bank’s asset is higher than that of its liability showing that the bank makes profit

International Journal of basic science and technology, 2017

The effect of death rate in determining the optimal investment strategies for defined contributio... more The effect of death rate in determining the optimal investment strategies for defined contribution (DC) pension fund with multiple contributors was investigated using a modified model. We assume a case where the wealth of death pensioners is not added to the pension wealth and also when their wealth is added to pension wealth. Using this model, we obtained optimized problems for the two assumptions using Jacobi Hamilton equation and solve the problems using Legendre transform to obtain an explicit solution of the optimal investment strategies for CARA utility function. We observed that the optimal investment strategies with the death pensioners' wealth is greater compared to one without their wealth. This model has shown that the wealth of the death pensioners has an effect on the overall investment strategies hence the pension manager makes more interest with the surplus from the death pensioners' wealth and loses more if the investment fails.

Journal of Mathematical Association of Nigeria, 2021

In this research paper, a defined contribution (DC) pension plan member's optimal portfolio strat... more In this research paper, a defined contribution (DC) pension plan member's optimal portfolio strategy with return of contribution clause under modified constant elasticity of variance (M-CEV) is studied. Considering investment in a risk-free asset and a risky asset modeled by a M-CEV process, a continuous time mean-variance stochastic optimal control problem consisting of members' monthly contributions, returned contributions and invested funds is formulated. Using the game theoretic method and mean variance utility, a non-linear partial differential equation (PDE) called the extended Hamilton Jacobi Bellman (HJB) is established and solved for the optimal portfolio strategy and efficient frontier using change of variable and variable separation technique. Also, theoretical analyses of the impact of the modification parameter and some other sensitive parameters on the optimal portfolio strategy were studied. Moreover, our result generalizes some existing results.

Nigerian Journal of Mathematics and Application, 2021

One of the major problems encountered by most insurance companies is portfolio management and pay... more One of the major problems encountered by most insurance companies is portfolio management and payment of claims. Hence the study of optimal portfolio strategy (OPS) and optimal reinsurance strategy (ORS) becomes necessary. In this paper, the insurer is allowed to invest in a risk-free-asset and a risky-asset, where the risky-asset price follows the constant elasticity variance model and can buy proportional reinsurance policy as a backup. By optimal control approach, the Hamilton-Jacobi-Bellman (HJB) equation is obtained. Using Legendre transformation method, the HJB-equation is transformed to a linear partial differential equation and solved for OPS and ORS for an insurer with logarithm utility. Finally, numerical and theoretical analyses were presented to study the impact of model parameters on ORS and OPS.

International journal of advances in mathematics, 2017

We investigate the optimal investment strategies of an insurance company. We assume that the rate... more We investigate the optimal investment strategies of an insurance company. We assume that the rates at which premiums are paid to insurance companies are stochastic, the total claims are modeled by a compound Poisson process, we assume that surplus of the insurance company is invested in risk free asset and in a risky asset such as stocks. We applied the Jacobi Hamilton-Jacobi-Bellman equation to obtain an optimized problem and used the Legendre transform and dual theory to obtain the optimal investment strategy for constant absolute risk aversion (CARA) utility function. Our result will enables insurance companies to determine the proportion of their wealth to be invested in risk-free asset and a risky asset in order to optimize profit knowing full well it has responsibility to pay the policy holders whenever there is claims occurrence.

International Journal of computer science and Mathematical theory, 2019

In this paper, we investigate the optimal strategy for a pension member in a defined contribution... more In this paper, we investigate the optimal strategy for a pension member in a defined contribution pension scheme under a market with inflation and minimum guarantee. We assume the contribution process includes the mandatory contribution and a supplementary contribution to amortize the pension fund which is assumed to be stochastic. Next, the management of the pension considers investments in cash, stock and inflation-linked bond to maximize the expected return of his member at the time of retirement. Using stochastic optimal control method, we derived an optimized problem from the Hamilton Jacobi Bellman (HJB) equation for the value function. Furthermore, we obtain the closed form solution of the optimal strategy for the three assets using constant absolute risk aversion (CARA) utility function and observed that the supplementary contributions has a direct effect on the inflation-linked bond and cash only. Keyword: DC Pension scheme, HJB, optimal investment strategy, inflation, supp...

This study centres on determining the optimal investment strategies for defined contribution (DC)... more This study centres on determining the optimal investment strategies for defined contribution (DC) pension fund with multiple contributors, administration cost and taxation on the invested fund. We assume that a certain proportion of the member’s contributions as administrative cost which is remitted to the pension fund manager also following the Nigerian Pension Reform Act of 2004 the invested fund is subjected to tax. We obtained an optimized equation using Hamilton Jacobi equation, then solve the equation using Legendre transformation method to obtained explicit solutions of the optimal investment strategy for CARA utility function. We observed that the tax has a direct effect on the investment strategies.