Adv Syst Sci Appl 2019; 03; 52-64 Published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/713 Optimal Values in an Uncertain Optimal Control Model with Application to Capital Asset Management Tolulope Latunde1* 1) Department of Mathematics, Federal University, Oye-Ekiti, Nigeria E-mail: tolulope.latunde@fuoye.edu.ng Received March 21, 2019; Revised September 6, 2019; Published October 1, 2019 Abstract: A new model of capital asset management was developed under the assumptions of hyperbolic absolute risk aversion, and employing the basic skills of mathematical modelling. The solution to the model was sought by formulating a continuous-time utility portfolio model satisfying some uncertainty criteria where the investment is continuous, the investor does not possess enough power to determine price and the investor can borrow money for a given period at a particular interest rate. The model was solved using analytical method and numerical method and optimal values of some input factors are derived. Keywords: Capital asset management, optimal control, uncertainty theory, optimal values. 1. INTRODUCTION Capital asset models have been subject to an enormous number of empirical studies since Lintner and Sharpe in the mid-1960. Among the most notable early tests of the models are those in [1, 4]. Other contributors are included in [3, 5, 13, 14]. The earliest researchers found the relationships between the risk-free rate and market risk premium. Since Merton developed and solve a portfolio selection model under uncertainty for the case of infinite lifetimes and finite lifetimes: the continuous-time case for risk and risk-free assets, capital asset management has adapted the portfolio theories, see [12]. This results to some researchers formulating asset management models using different mathematical approaches such that optimal values of controls and input factors are obtained. A stochastic optimal control approach was utilised to model debt crisis so as to evaluate debt crisis in international finance, thus, the optimal debt to preclude crisis was obtained [15]. The work [19] was based on the concepts of uncertainty theory where uncertain optimal control was applied to solve a portfolio selection model and obtained a fundamental result called equation of optimality for uncertain optimal control, thus obtain the optimal value of the control. Formulation of two instances of uncertain multidimensional optimal control models with 𝑛 jumps based on uncertainty theory from the perspectives of agents ( consumers) and principal (government) was carried out in the work [2] - the optimal control problems were applied to Research and Development fiscal policy and optimal control decisions were obtained. It is understood from existing works that the research carried out in [12,19, 2] examined the selection of risk-free asset and risk asset together in the formulation of the models. Meanwhile, it was examined in [15], the selection of risky assets only with stochastic optimal control approach without considering depreciation and taxation as input factors. The choice of Uncertainty theory over the conventional probability theory exists when the sample size is * Corresponding author: tolulope.latunde@fuoye.edu.ng mailto:tolulope.latunde@fuoye.edu.ng T. LATUNDE 53 Copyright Β©2019 ASSA. Adv. in Systems Science and Appl. (2019) small to estimate a probability distribution and degree belief are ascertained from experts to work in place of frequency since human beings always overweight unlikely events Consequently, a new model of asset management based on uncertainty theory was formulated in [6] where the selection of capital assets, classified as risky assets is examined. Thus, depreciation and taxation are considered as input factors in the formulation of the models. However, this work is an extension of the work [6, 7] such that we seek to provide solutions to a problem of capital assets using a real life situation. The proposed model deals with a case where a nation invests her wealth in capital assets, for a particular period of time. It is assumed that there is difficulty in deciding the fraction of the nation’s net worth to be incurred on the investments of capital assets, thus leading to the problem of how to optimize the expected present value of the utility of assets. 2. PRELIMINARY Uncertainty theory is a branch of mathematics for modelling belief degrees established by Liu, [8] and refined in [11]. The choice of Uncertainty theory over the conventional probability theory exists when the sample size is small to estimate a probability distribution and degree belief are ascertained from experts to work in place of frequency since human beings always overweight unlikely events. For the sake of this work, the following concepts are utilized. Let Ξ“ be a nonempty set and 𝐿 a 𝜎- algebra over Ξ“ such that (Ξ“, 𝐿) be a measurable space. Each element Ξ› ∈ 𝐿 is called an event. Definition 2.1 [8]: A set function 𝑀 defined on the 𝜎-algebra over 𝐿 is called an uncertain measure if it satisfies the following axioms: Axiom 1. (Normality Axiom): 𝑀{Ξ›} = 1 for the universal set 𝛀. Axiom 2. (Duality Axiom): 𝑀{Ξ›} + 𝑀{Λ𝑐} = 1 for any event Ξ›. Axiom 3. (Subadditivity Axiom): For every countable sequence of events, Ξ›1, Ξ›2, β‹―, we have 𝑀{β‹ƒβˆž 𝑖=1 Λ𝑖} ≀ βˆ‘βˆž 𝑖=1 𝑀{Λ𝑖} (2.1) Axiom 4. (Product Axiom): Let (Ξ“π‘˜, πΏπ‘˜, π‘€π‘˜) be uncertainty spaces for π‘˜ = 1,2, β‹―The product uncertain measure 𝑀 is an uncertain measure satisfying 𝑀{∏∞ π‘˜=1 Ξ›π‘˜} = min1β‰€π‘˜β‰€βˆžπ‘€π‘˜{Ξ›π‘˜} (2.2) where Ξ›π‘˜ are arbitrarily chosen events from πΏπ‘˜ for π‘˜ = 1,2, β‹―, respectively. Definition 2.2 [10]: An uncertain process 𝐢𝜎 is said to be a canonical Liu process if (i) 𝐢0 = 0 and almost all sample paths are Lipschitz continuous, (ii) 𝐢𝜎 has stationary and independent increments, (iii) every increment 𝐢𝑠+𝜎 βˆ’ 𝐢𝑠 is a normal uncertain variable with expected value 0 and variance 𝜎2. The uncertainty distribution of 𝐢𝜎 is Φ𝜎(π‘₯) = [1 + exp ( βˆ’πœ‹π‘₯ √3𝜎 )] βˆ’1 , π‘₯ ∈ β„œ (2.3) and the inverse distribution is Φ𝜎 βˆ’1(𝑦) = 𝜎√3 πœ‹ ln 𝑦 1βˆ’π‘¦ , 𝑦 ∈ β„œ (2.4) Definition 2.3 [8]: Let πœ‰ be an uncertain variable. Then the expected value of πœ‰ is defined by 𝐸[πœ‰] = ∫ +∞ 0 𝑀{πœ‰ β‰₯ π‘₯}𝑑π‘₯ βˆ’ ∫ 0 βˆ’βˆž 𝑀{πœ‰ ≀ π‘₯}𝑑π‘₯ (2.5) provided that at least one of the two integrals is finite Definition 2.4 [9]: An uncertain process 𝑋𝑑 is said to have independent increments if 𝑋𝑑1 βˆ’ 𝑋𝑑0 , 𝑋𝑑2 βˆ’ 𝑋𝑑1 , β‹― , π‘‹π‘‘π‘˜ βˆ’ π‘‹π‘‘π‘˜βˆ’1 54 OPTIMAL VALUES IN UNCERTIN OPTIMAL CONTROL MODEL Copyright Β©2019 ASSA Adv. in Systems Science and Appl. (2019) are independent uncertain variables where 𝑑1, 𝑑2, β‹― , π‘‘π‘˜ are any times with 𝑑0 < 𝑑1 < β‹― < π‘‘π‘˜ That is, an independent increment process means that its increments are independent uncertain variables whenever the time intervals do not overlap. It is noted that the increments are also independent of the initial state. Definition 2.5 [9]: Suppose 𝐢𝑑 is a canonical Liu process, and 𝑓 and 𝑔 are two functions. Then 𝑑𝑋𝑑 = 𝑓(𝑑, 𝑋𝑑)𝑑𝑑 + 𝑔(𝑑, 𝑋𝑑)𝑑𝐢𝑑 (2.6) is called an uncertain differential equation. A solution is a Liu process 𝑋𝑑 that satisfies (2.3) and (2.4) identically in 𝑑. Definition 2.6 [9]: Let 𝑋𝑑 be an uncertain process. Then for each 𝛾 ∈ 𝛀, the function 𝑋𝑑(𝛾)is called a sample path of 𝑋𝑑. Definition 2.7 [11]: An uncertain process 𝑋𝑑 is said to be sample-continuous if almost all sample paths are continuous functions with respect to time 𝑑. Definition 2.8 Uncertainty Distribution of Solution [16]: Let 𝛼 be a number with 0 < 𝛼 < 1. An uncertain differential equation 𝑑𝑋(𝑑) = 𝑓(𝑑, 𝑋(𝑑))𝑑𝑑 + 𝑔(𝑑, 𝑋(𝑑))𝑑𝐢(𝑑) is said to have an 𝛼-path 𝑋(𝑑)𝛼 if it solves the corresponding ordinary differential equation 𝑑𝑋(𝑑)𝛼 = 𝑓(𝑑, 𝑋(𝑑)𝛼)𝑑𝑑 + |𝑔(𝑑, 𝑋(𝑑))|Ξ¦βˆ’1(𝛼)𝑑𝑑 (2.7) where π›·βˆ’1(𝛼) is the inverse uncertainty distribution of standard normal uncertain variable, that is, Ξ¦βˆ’1(𝛼) = √3 πœ‹ ln 𝛼 1 βˆ’ 𝛼 , 𝛼 ∈ β„œ Theorem 2.1 Extreme Value of Solution [18]: Let 𝑋(𝑑) and 𝑋(𝑑)𝛼 be the solution and 𝛼-path of the uncertain differential equation (2.6). Then, for any time 𝑑 > 0 and strictly increasing function 𝐽(π‘₯), the supremum sup𝑑0≀𝑑≀𝑑𝑛 𝐽(𝑋(𝑑)) has an inverse uncertainty distribution Ξ¨βˆ’1(𝛼) = sup𝑑0≀𝑑≀𝑑𝑛 𝐽(𝑋(𝑑)𝛼) (2.8) and the infimum inf𝑑0≀𝑑≀𝑑𝑛 𝐽(𝑋(𝑑)) has an inverse uncertainty distribution Ξ¨βˆ’1(𝛼) = inf𝑑0≀𝑑≀𝑑𝑛 𝐽(𝑋(𝑑)𝛼) (2.9) Theorem 2.2 [16]: Let 𝑋(𝑑) and 𝑋(𝑑)𝛼 be the solution and 𝛼-path of the uncertain differential equation (2.6). Then 𝑀{𝑋(𝑑) ≀ 𝑋(𝑑)𝛼, βˆ€π‘‘} = 𝛼, 𝑀{𝑋(𝑑) > 𝑋(𝑑)𝛼, βˆ€π‘‘} = 1 βˆ’ 𝛼. (2.10) 3. MODEL FORMULATION A model of capital asset management is presented herein such that it is assumed that an investor invests his wealth in capital asset, 𝐴(𝑑), of a large business for time, 𝑑, from 𝑑0 to 𝑑𝑓. T. LATUNDE 55 Copyright Β©2019 ASSA. Adv. in Systems Science and Appl. (2019) Suppose he starts with a known initial net worth 𝑋0(𝑑). At time 𝑑, what ratio of his net worth, πœ“, must he select to utilize on capital asset in the presence of liability such that the expected net present value of the utility of asset, 𝐽(πœ“), is optimized ? Table 3.1. Definition of Parameters of the Objective function to the model Parameter Description π‘ˆ Utility function 𝐴(𝑑) Capital asset at time 𝑑 πœ‚ subjective discount rate, e.g., 𝐴 πœ‚+1 = Presentvalue πœ† degree of relative risk, where (1 βˆ’ πœ†) is the risk aversion πœ“ Capital asset ratio (control) πœ“ ∈ β„œ 𝑋(𝑑) Net worth at time (state variable)𝑑 Table 3.2: Definition of Parameters of the constraint to the model Parameter Description πœŽπ‘Ÿ(𝑑) Diffusion volatility of liability (with variance πœŽπ‘Ÿ 2 per unit time) πœ“ Capital asset ratio (control) πœ“ ∈ β„œ πœŽπ‘(𝑑) Diffusion volatility of asset (with variance πœŽπ‘ 2 per unit time) πœ…(𝑑) Capital gain on asset due to inflation at time 𝑑 πœŽπ‘(𝑑) Diffusion volatility on asset price (with variance πœŽπ‘ 2 per unit time) 𝛽(𝑑) Mean rate of return on asset πœ” Mean interest rate of liability 𝐢(𝑑) uncertain process at time 𝑑 πœ‡(𝑑) Consumption level at time 𝑑 𝑗(𝑑) Tax ratio at time 𝑑 𝑔(𝑑) Depreciation ratio at time 𝑑 β„Ž(𝑑) Asset supplies ratio at time 𝑑 Theorem 3.1 [12]: If πœ† > 0 and π‘ˆ is such that the integral 𝐸𝐢 [∫ 𝑑𝑓 𝑑0 π‘ˆ(𝐴, 𝑑)𝑑𝑑] is absolutely convergent, then the maximization or minimization of 𝐸𝐢 [∫ 𝑑𝑓 𝑑0 π‘ˆ(𝐴, 𝑑)𝑑𝑑] is equivalent to the maximization or minimization of 𝐸0 ∫ 𝑑𝑓 𝑑0 π‘’βˆ’πœ‚π‘‘π‘ˆ(𝐴, 𝑑)𝑑𝑑 where 𝐸𝐢 is the conditional expectation operator over all random variables excluding πœ†. By Theorem 3.1, an investor who faces an exponentially-distributed uncertain investment of capital asset invests as if there is no terminal period, but with a subjective rate of time preference equal to the investments terminals. 𝐽 = opt𝐸𝐢 [ ∫ 𝑑𝑓 𝑑0 π‘’βˆ’πœ‚π‘‘π‘ˆ(𝐴(𝑑))𝑑𝑑] (3.1) where 𝐸𝐢 denotes conditional expectation, πœ‚ ∈ (0,1) is the arbitrary discount rate. In selecting the discount rate, the effective length of time is inversely proportional to the discount rate in the sense that a high discount rate implies a short time interval, [15]. Utility function π‘ˆ(𝐴) measures satisfaction of an investor as a function of usage or efficiency of capital assets with respect to risk aversion. However, risk aversion is the behaviour of the investors when exposed 56 OPTIMAL VALUES IN UNCERTIN OPTIMAL CONTROL MODEL Copyright Β©2019 ASSA Adv. in Systems Science and Appl. (2019) to attempt to reduce the uncertainty in their investments. There are various measures of risk aversion under expected utility theory. Thus, a special case of Hyperbolic absolute risk aversion (HARA) is considered as the model’s utility function which helps in focusing more on ratios and its assumption also lowers the dimension of dynamic system for the model to be effortlessly solved analytically unlike some other utility functions. That is π‘ˆ(𝐴) = ( 1 πœ† π΄πœ†, 0 < πœ† < 1 lnA, πœ† = 0 (3.2) where 1 βˆ’ πœ† > 0 is the investor’s relative risk aversion. The larger the πœ†, the more reluctant to own a risky asset, [12]. The efficiency or performance of the capital asset, πœ“, is expressed as the ratio of capital asset, 𝐴(𝑑), and net worth, 𝑋(𝑑), such that 𝑋(𝑑) β‰  0, [15]. That is, πœ“ = 𝐴(𝑑) 𝑋(𝑑) (3.3) 𝐴(𝑑) = πœ“π‘‹(𝑑) From the proceeding, the model of risky capital asset is an optimal control of the form. 𝐽(πœ“) = maxπœ“πΈπΆ [ ∫ 𝑑𝑛 𝑑0 1 πœ† π‘’βˆ’πœ‚π‘‘(πœ“π‘‹(𝑑))πœ†π‘‘π‘‘] (3.4) subject to 𝑑𝑋(𝑑) = [(πœ… + 𝛽)πœ“ βˆ’ (πœ”(πœ“ βˆ’ 1) + πœ‡ + β„Ž βˆ’ 𝑗 βˆ’ 𝑔)]𝑋(𝑑)𝑑𝑑 +[πœ“πœŽπ‘ + πœ“πœŽπ‘ βˆ’ πœŽπ‘Ÿ(πœ“ βˆ’ 1)]𝑋(𝑑)𝑑𝐢(𝑑) [6] (3.5) 4. APPLICATION OF THE CAPITAL ASSET MODEL Here, the model is analysed using real life data in order to provide some optimal solutions satisfying the optimality criteria. Utilizing the model in international finance, the revenue is taken to be the Gross Domestic Product of a nation (GDP) or value added, the risky capital asset as the capital investment or the Gross Fixed Capital Formation, Consumption as Household and Government Consumption, Depreciation is taken as the Consumption of Fixed capital. The debt is also taken as a study case of liability to be considered. Table 4.1: Definition of parameters according to the model application Parameter Description 𝐽 Expected present value of utility of GFCF T. LATUNDE 57 Copyright Β©2019 ASSA. Adv. in Systems Science and Appl. (2019) πœ‚ subjective discount rate, e.g., 𝐴 πœ‚+1 = Presentvalue πœ† degree of relative risk, where (1 βˆ’ πœ†) is the risk aversion πœ“ GFCF ratio (control) πœ“ ∈ β„œ 𝜏(𝑑) Debt ratio (control) at time 𝑑, 𝜏 = πœ“ βˆ’ 1 𝑋(𝑑) Net worth at time 𝑑 (GFCF minus Debt ) πœ…(𝑑) Capital gain on GFCF with net worth at time 𝑑 𝛽(𝑑) Mean rate of return on GFCF with net worth πœ”(𝑑) Mean rate of debt with net worth πœŽπ‘(𝑑) Diffusion volatility on asset price (with variance πœŽπ‘ 2 per unit time) πœŽπ‘(𝑑) Diffusion volatility of GFCF (with variance πœŽπ‘ 2 per unit time) πœŽπ‘Ÿ(𝑑) Diffusion volatility of Debt (with variance πœŽπ‘Ÿ 2 per unit time) 𝜎(𝑑) Diffusion volatility of the whole process (πœŽπ‘ + πœŽπ‘ βˆ’ πœŽπœ”), [15] πœ‡(𝑑) Consumption level with net worth at time 𝑑 𝑠(𝑑) Net foreign supplies - net worth ratio at time 𝑑 𝑗(𝑑) Tax-net worth ratio at time 𝑑 𝑔(𝑑) Depreciation-net worth ratio at time 𝑑 𝐢(𝑑) Liu canonical process at time 𝑑 In order to examine the debt crisis in Nigeria and propose a warning signal, the data that are available after the Paris Debt forgiveness in 2006 are used. Thus, Tables 4.2 and 4.3 below represent the base parameter set for the case study. Table 4.2: Nigeria Net worth Profile Year GDP Debt GFCF Net worth Consumption Indirect Tax Depreciation Supplies 2007 166.451 22.330 15.396 -6.934 149. 152 2.553 3.738 5.089 2008 208.065 21.399 17.318 -4.081 161.035 3.436 3.853 30.988 2009 169.481 25.817 20.487 -5.330 147.601 3.180 2.952 0.445 2010 369.062 40.100 61.099 21.860 293.507 5.623 16.079 28.662 2011 411.744 47.898 63.960 16.062 323.540 4.516 18.815 38.719 2012 460.953 48.496 65.283 16.787 348.597 5.686 24.260 86.210 2013 514.966 64.510 72.964 8.454 453.699 7.929 23.857 26.280 2014 568.499 67.726 85.737 18.011 464.696 6.857 25.272 32.499 2015 481.066 65.429 71.329 5.900 417.560 5.362 23.097 0.000 2016 405.083 57.392 73.261 15.869 206.414 3.188 10.332 -3.341 Source: Columns 1 and 3. The world bank (http://data.worldbank.org/indicator) Column 2. Debt Management Office of Nigeria (https://www.dmo.gov.ng/). Columns 5, 6, 7, 8 and 9. National Bureau of Statistics (http://nigerianstat.gov.ng/) Table 4.3 is derived from Table 4.2. Table 4.3: Parameters for the Nigeria Net worth Profile Year πœ… 𝛽 πœ” πœŽπ‘ πœŽπ‘ πœŽπ‘Ÿ 𝜎 πœ‡ h j g 58 OPTIMAL VALUES IN UNCERTIN OPTIMAL CONTROL MODEL Copyright Β©2019 ASSA Adv. in Systems Science and Appl. (2019) 2007 -7.25 -51.72 -6.10 -3.78 -21.47 -2.46 -22.79 -21.51 -0.73 -0.37 -0.54 2008 -12.32 -87.88 -10.36 -6.42 -36.48 -4.17 -38.73 -39.46 -7.59 -0.77 -0.94 2009 -9.43 -67.29 -7.93 -4.92 -27.93 -3.20 -29.65 -27.69 0.08 -0.60 -0.55 2010 2.30 16.41 1.93 1.20 6.81 0.78 7.23 13.43 1.31 0.26 0.74 2011 3.13 22.33 2.63 1.63 9.27 1.06 9.84 20.14 2.14 0.28 1.17 2012 2.30 21.37 2.52 1.56 8.87 1.02 9.41 20.77 5.14 0.34 1.45 2013 5.95 42.42 5.00 3.10 17.61 2.02 18.69 53.67 3.11 0.94 2.82 2014 2.79 19.91 1.43 1.46 8.27 0.95 8.78 25.80 1.92 0.37 1.40 2015 8.52 60.79 4.37 4.44 25.23 2.89 26.78 70.77 0.00 0.91 3.91 2016 3.17 22.60 1.63 1.65 9.38 1.07 9.96 13.01 -0.21 0.20 0.65 Measurements The measurements considered in obtaining data in Tables 4.1 – 4.3 are described below. All the values of parameters are measured in Billion US Dollars except the following parameters: 𝐢(𝑑) - measures the uncertainty process which exists in the interval 0 < 𝐢(𝑑) < 1; πœ† - is used to measure risk which exists in the interval 0 < πœ† < 1; and πœ‚ - measures discount rate which exists in the interval 0 < πœ‚ < 1. The debt is calculated as the total debt of the nation by summing the external debt stock (federal government and state) and Domestic debt (federal government and state) together. The net worth is also calculated by deducting the debt from the GFCF. The CBN Official Exchange rate of π‘ˆπ‘†π· at 31st December of each year is used while current market prices from the national account are used in the computations. 4.1 Solution to the model Here, the analytical and numerical solutions are derived. For the analytic solution, the required problem under consideration is 𝐽(πœ“) = minπœ“πΈπΆ [ ∫ 𝑑𝑛 𝑑0 1 πœ† π‘’βˆ’πœ‚π‘‘(πœ“π‘‹(𝑑))πœ†π‘‘π‘‘] subject to 𝑑𝑋(𝑑) = [(πœ… + 𝛽)πœ“ βˆ’ (πœ”(πœ“ βˆ’ 1) + πœ‡ + 𝑠 βˆ’ 𝑗 βˆ’ 𝑔)]𝑋(𝑑)𝑑𝑑 +[πœ“πœŽπ‘ + πœ“πœŽπ‘ βˆ’ πœŽπ‘Ÿ(πœ“ βˆ’ 1)]𝑋(𝑑)𝑑𝐢(𝑑) with 𝛼-path equation 𝑑𝑋(𝑑)𝛼 = [(πœ… + 𝛽)πœ“ βˆ’ (πœ”(πœ“ βˆ’ 1) + πœ‡ + β„Ž βˆ’ 𝑗 βˆ’ 𝑔)]𝑋(𝑑)𝛼𝑑𝑑 + |[πœ“πœŽπ‘ + πœ“πœŽπ‘ βˆ’ πœŽπ‘Ÿ(πœ“ βˆ’ 1)]𝑋(𝑑)𝛼|Ξ¦βˆ’1(𝛼)𝑑𝑑. The analytical solution to the constraint is T. LATUNDE 59 Copyright Β©2019 ASSA. Adv. in Systems Science and Appl. (2019) 𝑋(𝑑) = 𝑋0exp([(πœ… + 𝛽)πœ“ βˆ’ (πœ”(πœ“ βˆ’ 1) + πœ‡ + 𝑠 βˆ’ 𝑗 βˆ’ 𝑔)]𝑑 + [πœ“πœŽπ‘ + πœ“πœŽπ‘ βˆ’ πœŽπ‘Ÿ(πœ“ βˆ’ 1)]𝐢(𝑑)) and its inverse uncertainty distribution is Ξ¨(𝑑)βˆ’1(𝛼) = 𝑋0exp([(πœ… + 𝛽)πœ“ βˆ’ (πœ”(πœ“ βˆ’ 1) + πœ‡ + 𝑠 βˆ’ 𝑗 βˆ’ 𝑔)]𝑑 + [πœ“πœŽπ‘ + πœ“πœŽπ‘ βˆ’ πœŽπ‘Ÿ(πœ“ βˆ’ 1)]π‘‘βˆš3 πœ‹ ln 𝛼 1 βˆ’ 𝛼 ) Hence, by Theorem 2.1, Ξ¨(𝑑)βˆ’1(𝛼) = 𝐸(𝑋(𝑑)𝛼) Numerical solutions are presented via trapezoidal rule for the objective functional and, Euler method and fourth order Runge-Kutta method for solving uncertain differential equations due to its ability to yield more precise outcomes than other methods for the constraints, [17]. Trapezoidal method: ∫ 𝑏 π‘Ž 𝑓(π‘₯)𝑑π‘₯ = β„Ž 2 (𝑓0 + 2𝑓1 + 2𝑓2 + β‹― + 2π‘“π‘›βˆ’1 + 𝑓𝑛) = β„Ž( 𝑓0 + 𝑓𝑛 2 + βˆ‘ π‘›βˆ’1 𝑖=1 𝑓𝑖), where 𝑓(π‘₯π‘˜) ≑ π‘“π‘˜ The Runge-Kutta method for solving uncertain differential equations was designed in [17] with respect to the following definition and theorems. Runge-Kutta method is an effective method for solving ordinary differential equations. The generally used Runge-Kutta formula is a fourth-order formula. It should be noted that there is a wide range of fourth-order schemes and here, just one common structure is exhibited. For an ordinary differential equation with initial value 𝑋0 𝑑𝑋(𝑑) = 𝐹(𝑑, 𝑋(𝑑))𝑑𝑑. The scheme uses the following formula 𝑋(𝑑𝑛+1) = 𝑋(𝑑𝑛) + 1 6 (π‘˜1 + 2π‘˜2 + 2π‘˜3 + π‘˜4) where π‘˜1 = β„ŽπΉ(𝑑𝑛, 𝑋𝑛), π‘˜2 = β„ŽπΉ(𝑑𝑛 + β„Ž 2 , 𝑋𝑛 + 1 2 π‘˜1), π‘˜3 = β„ŽπΉ(𝑑𝑛 + β„Ž 2 , 𝑋𝑛 + 1 2 π‘˜2), π‘˜4 = β„ŽπΉ(𝑑𝑛 + β„Ž, 𝑋𝑛 + π‘˜3) and β„Ž is the step size which is assumed to be constant for all steps. However, based on Theorem 2.4, a Runge-Kutta method for uncertain differential equations was designed as 𝑋𝑖+1 𝛼 = 𝑋𝑖 𝛼 + 1 6 (π‘˜1 + 2π‘˜2 + 2π‘˜3 + π‘˜4) where 60 OPTIMAL VALUES IN UNCERTIN OPTIMAL CONTROL MODEL Copyright Β©2019 ASSA Adv. in Systems Science and Appl. (2019) π‘˜1 = β„Ž(𝑓(𝑑𝑖, 𝑋𝑖 𝛼) + |𝑔(𝑑𝑖, 𝑋𝑖 𝛼) |Ξ¦βˆ’1(𝛼)), π‘˜2 = β„Ž(𝑓(𝑑𝑖 + β„Ž 2 , 𝑋𝑖 𝛼 + 1 2 π‘˜1) + |𝑔(𝑑𝑖 + β„Ž 2 , 𝑋𝑖 𝛼 + 1 2 π‘˜1)|Ξ¦βˆ’1(𝛼)), π‘˜3 = β„Ž(𝑓(𝑑𝑖 + β„Ž 2 , 𝑋𝑖 𝛼 + 1 2 π‘˜2) + |𝑔(𝑑𝑖 + β„Ž 2 , 𝑋𝑖 𝛼 + 1 2 π‘˜2)|Ξ¦βˆ’1(𝛼)), π‘˜4 = β„Ž(𝑓(𝑑𝑖 + β„Ž, 𝑋𝑖 𝛼 + π‘˜3) + |𝑔(𝑑𝑖 + β„Ž, 𝑋𝑖 𝛼 + π‘˜3)|Ξ¦βˆ’1(𝛼)). For the proposed optimal control model of net risky capital asset with an uncertain differential equation 𝑑𝑋(𝑑) = [(πœ… + 𝛽)πœ“ βˆ’ (πœ”(πœ“ βˆ’ 1) + πœ‡ + β„Ž βˆ’ 𝑗 βˆ’ 𝑔)]𝑋(𝑑)𝑑𝑑 + [πœ“πœŽπ‘ + πœ“πœŽπ‘ βˆ’ πœŽπ‘Ÿ(πœ“ βˆ’ 1)]𝑋(𝑑)𝑑𝐢(𝑑) with initial value 𝑋0 and its 𝛼-path equation. i.e., 𝑑𝑋(𝑑)𝛼 = [(πœ… + 𝛽)πœ“ βˆ’ (πœ”(πœ“ βˆ’ 1) + πœ‡ + β„Ž βˆ’ 𝑗 βˆ’ 𝑔)]𝑋(𝑑)𝛼𝑑𝑑 +|[πœ“πœŽπ‘ + πœ“πœŽπ‘ βˆ’ πœŽπ‘Ÿ(πœ“ βˆ’ 1)]𝑋(𝑑)𝛼|Ξ¦βˆ’1(𝛼)𝑑𝑑, This is solved using the algorithm below. 4.2. Algorithm 4.1: Runge-Kutta method for solving the model Step 1. Given time interval 𝑑, [π‘Ž, 𝑏], iteration number 𝑁, step length β„Ž = π‘βˆ’π‘Ž 𝑁 . Set 𝑑𝑖 = π‘Ž + π‘–β„Ž, 𝑖 = 0,1, β‹― , 𝑁 and 𝛼 > 0. Step 2. Compute the corresponding differential equation 𝑑𝑋(𝑑)𝛼 = [(πœ… + 𝛽)πœ“ βˆ’ (πœ”(πœ“ βˆ’ 1) + πœ‡ + 𝑠 βˆ’ 𝑗 βˆ’ 𝑔)]𝑋(𝑑)𝛼𝑑𝑑 +|[πœ“πœŽπ‘ + πœ“πœŽπ‘ βˆ’ πœŽπ‘Ÿ(πœ“ βˆ’ 1)]𝑋(𝑑)𝛼| 𝜎√3 πœ‹ ln 𝛼 1βˆ’π›Ό 𝑑𝑑, 𝑋0 𝛼 = 𝑋0, with the Runge-Kutta method for solving uncertain differential equations. Step 3. Set 𝑖 = 𝑖 + 1, repeat Step 2 and step 3 for 𝑁 times, then 𝑋(𝑑)𝛼 is derived. Go back to step 1 until 𝑑𝑖 = 𝑏, Table 4.4: Results of Analytical solution to the Model with β„Ž = 0.05, π‘Ž ≀ 𝑑 ≀ 𝑏, π‘Ž = 0, 𝑏 = 1, πœ‚ = 0.9, πœ† = 0.1 and 𝑋0 = 18.011 πœ“ X J -9 -1175.785 16.657 -7 -28.783 11.209 -5 -0.277 6.812 -3 -0.118 5.941 -1 -1.612 6.916 1 19.050 8.854 3 95.377 11.609 5 91.492 12.166 7 77242.383 24.684 9 4.159 Γ— 106 37.710 Table 4.5: Results of Numerical solution to the Model (β„Ž = 0.05, π‘Ž = 0, 𝑏 = 1, πœ‚ = 0.9, πœ† = 0.1 and 𝑋0 = 18.011) πœ“ X J -9 -1638.001 17.218 -7 -31.576 11.313 T. LATUNDE 61 Copyright Β©2019 ASSA. Adv. in Systems Science and Appl. (2019) -5 -0.322 6.916 -3 -0.150 6.088 -1 -3.037 7.368 1 19.007 8.852 3 93.596 11.587 5 84.931 12.076 7 74991.214 24.612 9 3.524 Γ— 106 37.090 Furthermore, using the available data, the numerical and analytical solution to the asset-liability management problem was presented. Using the net worth of the year 2007 to 2016 were to study the behaviour of the optimal control in each year and provide a control policy to the problem. Let πœ“π‘‚ be the optimal control which implies the rate of capital asset such that the expected present value of the utility of assets, and πœ“π΄ be the actual control which is obtained from the given ratio of capital asset and net worth. The optimal control πœ“π‘‚ for each year was derived analytically using the equation of optimality proposed in [19] for uncertain optimal control problem, where πœ“π‘‚ = (πœ‡+𝑗+𝑔+πœ”βˆ’β„Ž)πœ†βˆ’πœ‚ (1βˆ’πœ†)(πœ…+π›½βˆ’πœ”) and the actual control is πœ“π΄ = 𝐴(𝑑) 𝑋(𝑑) . Table 4.6: Results on performance of Capital Asset Year πœ“π΄ πœ“π‘‚ πœ“π΄ - πœ“π‘‚ 2007 -2.22 -4.09 1.87 2008 -4.24 -5.88 1.64 2009 -3.84 -5.09 1.25 2010 2.80 0.67 2.13 2011 3.98 1.42 2.56 2012 3.89 1.22 2.67 2013 8.63 5.59 3.04 2014 4.76 2.01 2.75 2015 12.09 7.88 4.21 2016 4.62 0.74 3.88 The warning signal will be based on the difference between the actual liability ratio 𝜏𝐴 and the optimal liability ratio πœπ‘‚. Table 4.7: Warning signal Year 𝜏𝐴 πœπ‘‚ 𝜏𝐴 - πœπ‘‚ 2007 -3.22 -5.09 0.87 62 OPTIMAL VALUES IN UNCERTIN OPTIMAL CONTROL MODEL Copyright Β©2019 ASSA Adv. in Systems Science and Appl. (2019) 2008 -5.24 -6.88 0.64 2009 -4.84 -6.09 0.25 2010 1.80 -0.33 1.13 2011 2.98 0.42 1.56 2012 2.89 0.22 2.67 2013 7.63 4.59 2.04 2014 3.76 1.01 1.75 2015 11.09 6.88 3.21 2016 3.62 -0.26 2.88 4.2 Optimal U, 𝝀 and 𝜼 By hyperbolic absolute risk aversion utility function, π‘ˆ(𝐴) = ( 1 πœ† π΄πœ† , 0 < πœ† < 1 lnA, πœ† = 0 π‘‘π‘ˆ π‘‘πœ† = 0 βˆ’ 1 πœ†2 π΄πœ† + 1 πœ† π΄πœ†ln𝐴 = 0 1 πœ† = ln𝐴 πœ† = 1 ln𝐴 Hence, the optimal πœ† is πœ†βˆ— = 1 ln𝐴 . Similarly, optimal π‘ˆ is π‘ˆβˆ— where π‘ˆβˆ— = 1 πœ†βˆ— π΄πœ†βˆ— = (ln𝐴)𝐴 1 ln𝐴 From Table 4.2, Let 𝐴1 = 15.396 and 𝐴2 = 85.737 where πœ† = πœ†1 and 𝐴 = 𝐴1. This implies πœ†1 = 1 ln𝐴1 = 0.366 Let π‘ˆ = π‘ˆ1 π‘ˆ1 = (ln𝐴1)𝐴1 1 ln𝐴1 = 7.432 and let πœ† = πœ†2, 𝐴 = 𝐴2 πœ†2 = 1 ln𝐴2 = 0.225 π‘ˆ2 = (ln𝐴2)𝐴1 1 ln𝐴2 = 12.100 Therefore, πœ†2 ≑ πœ†βˆ— = 0.225 and π‘ˆ2 ≑ π‘ˆβˆ— = 12.100 Also, from equation (3.1) T. LATUNDE 63 Copyright Β©2019 ASSA. Adv. in Systems Science and Appl. (2019) 𝐽 = max𝐸𝐢 [ ∫ 𝑑𝑓 𝑑0 π‘’βˆ’πœ‚π‘‘π‘ˆ(𝐴(𝑑))𝑑𝑑] = maxπ‘ˆ(𝐴)max ∫ 1 0 π‘’βˆ’πœ‚π‘‘π‘‘π‘‘ = π‘ˆβˆ—max 1 πœ‚ (1 βˆ’ π‘’βˆ’πœ‚) = 1 πœ‚ π‘ˆβˆ— Given in Table 4.1 is the net present value 𝐽 as 𝐴 πœ‚+1 This implies 𝐴2 πœ‚ + 1 = π‘ˆβˆ— πœ‚ Thus, πœ‚βˆ— = π‘ˆβˆ— 𝐴2 βˆ’ π‘ˆβˆ— β‡’ πœ‚βˆ— = 0.164 Therefore, using the calculated optimal values, the following results are obtained. Table 4.8: Numerical result of Expected present value of utility of asset 𝐽(πœ“)𝛼 with different 𝛼-paths 𝛼 𝐽(πœ“)𝛼 10βˆ’6 1.152 Γ— 106 .1 25619.235 .24 6056.883 .38 448.529 .52 429.982 .64 3372.794 .8 15826.851 .999999 1.273 Γ— 106 5. CONCLUSION Based on uncertainty theory, an optimal control model of capital asset was formulated by adapting the expected value operator to quantify the objective rewards in the model. Furthermore, the model was solved by using some optimality criteria to derive the optimal values of some input factors, thus applied to a real life problem. 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