J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 Journal of the Nigerian Society of Physical Sciences An Investor’s Investment Plan with Stochastic Interest Rate under the CEV Model and the Ornstein-Uhlenbeck Process Edikan E. Akpanibaha,∗, Udeme O. Inib aDepartment of Mathematics and Statistics, Federal University Otuoke, Bayelsa, Nigeria bDepartment of Mathematics and Computer Science, Niger Delta University, Bayelsa, Nigeria Abstract The aim of this paper is to maximize an investor’s terminal wealth which exhibits constant relative risk aversion (CRRA). Considering the fluctuating nature of the stock market price, it is imperative for investors to study and develop an effective investment plan that considers the volatility of the stock market price and the fluctuation in interest rate. To achieve this, the optimal investment plan for an investor with logarithm utility under constant elasticity of variance (CEV) model in the presence of stochastic interest rate is considered. Also, a portfolio with one risk free asset and two risky assets is considered where the risk free interest rate follows the Ornstein-Uhlenbeck (O-U) process and the two risky assets follow the CEV process. Using the Legendre transformation and dual theory with asymptotic expansion technique, closed form solutions of the optimal investment plans are obtained. Furthermore, the impacts of some sensitive parameters on the optimal investment plans are analyzed numerically. We observed that the optimal investment plan for the three assets give a fluctuation effect, showing that the investor’s behaviour in his investment pattern changes at different time intervals due to some information available in the financial market such as the fluctuations in the risk free interest rate occasioned by the O-U process, appreciation rates of the risky assets prices and the volatility of the stock market price due to changes in the elasticity parameters. Also, the optimal investment plans for the risky assets are directly proportional to the elasticity parameters and inversely proportional to the risk free interest rate and does not depend on the risk averse coefficient. DOI:10.46481/jnsps.2021.172 Keywords: O-U process, Stochastic interest rate, Optimal investment plan, Legendre transform, CEV model Article History : Received: 13 March 2021 Received in revised form: 27 May 2021 Accepted for publication: 07 June 2021 Published: 29 August 2021 c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: T. Latunde 1. Introduction In the study of the optimal investment plan for any financial institution and considering the fact that the economy of most countries and even the financial markets are presently in seri- ous crisis due to the suffocating impact of the novel Covid-19 pandemic, the role of stochastic volatilities cannot be over em- phasized as it plays a significant role in the behaviour of the ∗Corresponding author tel. no: +2347030811189 Email address: edemae@fuotuoke.edu.ng (Edikan E. Akpanibah ) stock market price due to its unstable nature resulting from var- ious information obtainable from the market. In order to take a seemingly right decision while investing in risky assets such as stock, the stochastic volatility models become necessary to un- derstand the random nature of the stock market price. In this pa- per, we consider the CEV model which is one of the stochastic volatility models used in describing the behaviour of the stock market price. This model was first used in [2] and degenerate into a constant volatility model called the geometric Brownian motion (GBM) when its elasticity parameter equals zero. One significant property of this model is its ability to capture the 239 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 240 implied volatility skew. Several works have been done on utility maximization under the CEV model, some of which include [3] - [5]. In [6], the rein- surance problem and optimal investment under the CEV model was studied. [1], studied the optimal investment problem with taxes, dividend and transaction cost using different utility func- tions under the CEV process. [7, 8] solved the optimal invest- ment problem for a defined contribution (DC) pension plan with return of premiums clauses under different assumptions and as- sumed that the stock market price follows the CEV process; they used the game theoretic approach and solved the resultant extended Hamilton Jacobi Bellman equation for the optimal in- vestment plan. The optimal investment plan with stochastic in- terest rate under GBM has been studied by some authors; these include [9], who studied the investment portfolio under stochas- tic interest rate for a case of protected DC fund. Also, [10, 11], used stochastic interest rate to obtain optimal investment plan for a DC plan. In [12, 13], the optimal investment plan with the Vasicek interest rate was studied while [14] - [16] solved some optimization problems for the optimal investment plan when the interest rate is of affine type. The optimal investment plan with the CEV process under sev- eral assumptions has been studied by different authors when the risk free interest rate are constant as seen in the above litera- tures except for [17] - [19], who investigated the optimal control strategies under the CEV model with stochastic interest rate. In [17], the optimal investment plan of an insurer with stochastic interest rate under the CEV model was studied using logarithm utility, they assumed that the risk free interest rate follows the Cox- Ingersoll-Ross (CIR) process and used the power transfor- mation, change of variable and asymptotic approach to find the asymptotic solution of the optimal investment plan. In [18], the expected utility of an investor with exponential utility function was maximized using the Legendre transformation and asymp- totic expansion method to find a closed form solution of the optimal investment plan. They pointed out that authors find it difficult to combine the CEV model and stochastic interest rate in determining investment strategies due to the complexity of the resultant optimization problem. Also, they pointed out that in real life applications, interest rates are usually not constant but fluctuating in nature and the volatility of the interest rate generate some market risks; that is to say, when this risk are not considered, we are under estimating the effect of this risk emanating from this interest rate which is critical in influenc- ing the prices of different assets available in financial market. In [19], the optimal investment plan for an investor with expo- nential utility under the modified CEV model was studied. In their work, the risk free rate follows the O-U process. Most of the literatures mentioned above used the (CIR) model, Vasicek model, affine model etc. to model their interest rate, however very few used O-U process. According to [21], we know that it is more accurate to use O-U process in modelling both interest rate and stock market price since it reflects the fluctuation of the interest rates and asset prices. Also, the O-U process is closer to the change in interest rate. Based on this important characteristic of the O-U process and considering the unstable nature of the financial market at this present time, we are motivated to develop a robust investment plan for an investor with logarithm utility which takes into con- siderations fluctuations in interest rate as well as the volatility of the stock market prices. We do this by studying the opti- mal investment plans of an investor exhibiting the CRRA and whose risky assets and risk free interest rate are modelled by the CEV process and the O-U process respectively. Further- more, the Legendre transformation and asymptotic technique are used to determine asymptotic solutions of the optimal in- vestment plan. We also give some numerical simulations to explain our results. The main difference between our work and that of [18] is that we will consider an investor with logarithm utility where the investor’s behaviour toward risks are not influ- enced by the risk aversion coefficient instead of the exponential utility which depend on the risk aversion coefficient. Also, in- vestment is done with three assets instead of two assets and the risk free interest rate follows the O-U process instead of the CIR process. The rest of the paper is outlined as follows; Section 2 described the financial models of the three assets and the risk free rate. Section 3 gives the definition of the value function, derives the corresponding Hamilton Jacobi Bellman equations by maximum principle and application of Legendre transform to the HJB equation. Section 4 provides closed-form solutions of the optimal investment plan for the three assets under CRRA utility function. Section 5 and 6 present numerical simulations of the effects of some sensitive parameters on the optimal in- vestment plan and discussions. Section 7 concludes the work. 2. The Financial Market Model Consider a portfolio comprising of one risk free asset and two risky assets in a financial market which is open continuously for an interval t ∈ [0, T ] where T is the expiration date of the in- vestment. Let {B0 (t) , B1 (t) ,B2 (t) : t ≥ 0} be standard Brow- nian motion defined on a complete probability space (Ω, F, P) where Ω is a real space and P is a probability measure and F is the filtration which represents the information generated by the three Brownian motions. Let St (t) denote the price of the risk free asset at time t and the model is given as follows dS0 (t) S0 (t) = r(t)dtS0 (0) = s0 > 0 (1) where r(t) is the short interest rate process and driven by the stochastic differential equation dr (t) = θ(µ0 − r (t))dt −δdB0(t), r (0) = r0 > 0 (2) where θ, µ0 and δ are positive real numbers see [19] - [21]. Let S1 (t) and S2 (t) denote the prices of two different stocks which are described by the CEV model and the dynamics of the stock market prices are described by the stochastic differential equations at t ≥ 0 as follows dS1 (t) S1 (t) = µ1dt + σ11S β 1 (t) dB1 (t) + σ12S β 1(t)dB2(t) (3) dS2 (t) S2 (t) = µ2dt + σ21S β 2 (t) dB1 (t) + σ22S β 2(t)dB2(t) (4) 240 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 241 where µ1 and µ2 are appreciation rate of the two risky assets, σ11S β 1, σ12S β 1, σ21S β 2and σ22S β 2 are instantaneous volatilities and form a 2 × 2 matrix σ = {σmn}2×2 such that σσ T is positive definite and β < 0 represent elasticity parameter. (see [20] for details). Note if β = 0, the stock market price is modelled by GBM. 3. Optimization Problem 3.1. Wealth Formulations and HJB Equation Let ϕ be the optimal portfolio strategy and we define the utility attained by the investor from a given state x at time t as Lϕ (t, r, s1, s2, x) = (5) Eϕ [ U (X (T )) ∣∣∣∣∣∣ r (t) = r,S1 (t) = s1, S2 (t) = s2,X (t) = x ] , where t is the time, r is the risk free interest rate and x is the wealth. The objective here is to determine the optimal portfolio strate- gies ϕ∗ = (ϕ0, ϕ1, ϕ2) and the optimal value function of the investor L (t, r, s1, s2, x) given as L (t, r, s1, s2, x) = sup ϕ∗ Lϕ∗ (t, r, s1, s2, x) (6) such that Lϕ∗ (t, r, s1, s2, x) = L (t, r, s1, s2, x) . (7) Let X (t) be the investor’s wealth at time t and the differen- tial form associated with the fund size is given by dX (t) = X (t) ( ϕ0 dS0 (t) S0 (t) + ϕ1 dS1 (t) S1 (t) + ϕ2 dS2 (t) S2 (t) ) . (8) Substituting (1), (3) and (4) into (8), we have dX (t) = X (t)  (ϕ1 (µ1 − r) + ϕ2 (µ2 − r) + r) dt + ( ϕ1σ11S β 1 (t) + ϕ2σ21S β 2 (t) ) dB1 + ( ϕ1σ12S β 1 (t) + ϕ2σ22S β 2 (t) ) dB2 X (0) = X0  ,(9) where ϕ0, ϕ1 and ϕ2 are the optimal investment plans for the risk-free asset and the two risky assets respectively, such that ϕ0 = 1 −ϕ1 −ϕ2. From [19], applying the Ito’s lemma and maximum principle, the Hamilton Jacobi Bellman (HJB) equation which is a nonlin- ear PDE associated with (9) is obtained by maximizing Lϕ∗ (t, r, s1, s2, x) subject to the investor’s wealth as follows Lt + µ1 s1Ls1 + µ2 s2Ls2 + 1 2As 2β+2 1 Ls1 s1 + 12Bs 2β+2 2 Ls2 s2 + Cs β+1 1 s β+1 2 Ls1 s2 +rxLx + θ (µ0 − r)Lr + 12 δ 2Lrr + Dδρs β+1 1 Ls1 r + Eδρsβ+12 Ls2 r +supϕ1,ϕ2  ( 1 2Aϕ 2 1 s 2β 1 + Cϕ1ϕ2 s β 1 s β 2 + 12Bϕ 2 2 s 2β 2 ) Lxx ((µ1 − r) ϕ1 + (µ2 − r) ϕ2)Lx + ( Dδρϕ1 s β 1 + Eδρϕ2 s β 2 ) Lxr( As2β+11 ϕ1 + Cϕ2 s β+1 1 s β 2 ) Lxs1 + ( +Cϕ2 s β+1 1 s β 2 + Bϕ2 s 2β+1 2 ) Lxs2   = 0(10) Differentiating (10) with respect to ϕ1 and ϕ2, we obtain the first order maximizing condition for equation (10) as ϕ∗1 =  − [ Csβ1 (µ2−r)−Bs β 2 (µ1−r) ] x(AB−C2)s2β1 s β 2 Lx Lxx − s1 Lxs1 xLxx − (BD−CE)δρ x(AB−C2)sβ1 Lxr Lxx  (11) ϕ∗2 =  − [ Csβ2 (µ1−r)−As β 1 (µ2−r) ] x(AB−C2)sβ1 s 2β 2 Lx Lxx − s2 Lxs2 xLxx − (AE−CD)δρ x(AB−C2)sβ2 Lxr Lxx  (12) Substituting (11) and (12) into (10), we have Lt + µ1 s1Ls1 + µ2 s2Ls2 + 1 2As 2β+2 1 Ls1 s1 + 12Bs 2β+2 2 Ls2 s2 + Cs β+1 1 s β+1 2 Ls1 s2 +θ (µ0 − r)Lr + 1 2 δ 2 Jrr + Dδρs β+1 1 Ls1 r +Eδρsβ+12 Ls2 r + 1 2 ( F sβ1 s β 2 − G s2β1 − H s2β2 ) L2x Lxx +rxLx − (µ1 − r) s1 LxLxs1 Lxx − (µ2 − r) s2 LxLxs2 Lxx −δρ ( k1 (µ1−r) sβ1 + k2 (µ2−r) sβ2 ) LxLxr Lxx − 1 2As 2β+2 1 L2xs1 Lxx − 1 2Bs 2β+2 2 L2xs2 Lxx − 1 2 k3ρ 2δ2 L2xr Lxx −Csβ+11 s β+1 2 Lxs1Lxs2 Lxx  = 0(13) where A = σ211 + σ 2 12, B = σ 2 21 + σ 2 22, C = σ11σ21 + σ12σ22, D = σ11 + σ12,E = σ21 + σ22 F = 2C(µ1−r)(µ2−r) (AB−C2) ,G = B(µ1−r) 2 (AB−C2),H = A(µ2−r) 2 (AB−C2), k1 = (BD−CE) (AB−C2), k2 = (AE−CD) (AB−C2) .k3 = (BD2 +AE2−2CDE) (AB−C2) (14) From [20], we assumed that the optimal investment plan for risky assets’ prices are known based on the assumption that µ1ϕ ∗ 1 + µ2ϕ ∗ 2 = n (15) where n is a constant. Substituting (11) and (12) into (15), we derive an expression for 1 sβ1 s β 2 as 1 sβ1 s β 2 =  AB−C2 C(2µ1µ2−µ1 r−µ2 r) ×  nxLxx Lx + Bµ1 (µ1−r) (AB−C2)s2β1 + µ1 s1 Lxs1 Lx + (BD−CE)µ1δρ (AB−C2)sβ1 Lxr Lx + Aµ2 (µ2−r) (AB−C2)s2β2 +µ2 s2 Lxs2 Lx + (AE−CD)µ2δρ (AB−C2)sβ2 Lxr Lx   (16) Substituting (16) into (13), we have Lt + µ1 s1Ls1 + µ2 s2Ls2 + [r + α1] xLx + 12As 2β+2 1 Ls1 s1 + 1 2Bs 2β+2 2 Ls2 s2 +Csβ+11 s β+1 2 Ls1 s2 + 1 2 δ 2Lrr + Dδρs β+1 1 Ls1 r +Eδρsβ+12 Ls2 r + 1 2 ( α2 s −2β 1 +α3 s −2β 2 ) L2x Lxx +θ (µ0 − r)Lr + ( α4 − (µ1 − r) s1 ) LxLxs1 Lxx + ( α5 − (µ2 − r) s2 ) LxLxs2 Lxx + ( α6 s −β 1 +α7 s −β 2 ) LxLxr Lxx − 1 2As 2β+2 1 L2xs1 Lxx − 1 2Bs 2β+2 2 L2xs2 Lxx − 1 2 k3ρ 2δ2 L2xr Lxx −Csβ+11 s β+1 2 Lxs1Lxs2 Lxx  = 0(17) 241 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 242 α1 = F n(AB−C2) 2C(2µ1µ2 −µ1r −µ2r) ,α2 = FBµ1 (µ1 − r) (AB−C2) 2C(2µ1µ2 −µ1r −µ2r) −G, α3 = FAµ2 (µ2 − r) (AB−C2) 2C(2µ1µ2 −µ1r −µ2r) −H, α4 = Fµ1 s1 (µ2 − r) (AB−C2) 2C(2µ1µ2 −µ1r −µ2r) , α5 = Fµ2 s2 (µ2 − r) (AB−C2) 2C(2µ1µ2 −µ1r −µ2r) , α6 = (BD−CE)Fµ1δρ 2C(2µ1µ2 −µ1r −µ2r) − k1δρ (µ1 − r) , α7 = (AE−CD)Fµ1δρ 2C(2µ1µ2 −µ1r −µ2r) − k2δρ (µ2 − r) 3.2. Legendre Transformation and Dual theory The differential equation obtained in (17) is a non linear PDE and is somehow complex to solve. In this section, we will in- troduce the Legendre transformation and dual theory and use it to transform the non linear PDE to a linear PDE. Let f : Rn → R be a convex function for z > 0, define the Legendre transform N (z) = max x { f (x) − zx} , (18) The function N(z) is the Legendre dual of the function f (x). (c.f. [22]) Since f (x) is convex, from Theorem 3.2 and [21], the Legendre transform for the value function L (t, r, s1, s2, x) can be defined as follows L̂ (t, r, s1, s2, z) = (19) sup L (t, r, s1, s2, x) − zx | 0 < x < ∞} 0 < t < T where L̂ is the dual of L and z > 0 is the dual variable of x, The value of x where this optimum is achieved is represented by g (t, r, s1, s2, z) , such that g (t, r, s1, s2, z) = (20) inf x ∣∣∣∣ L (t, r, s1, s2, x) ≥ zx + L̂ (t, r, s1, s2, z)} 0 < t < T.  From (20), the function g and L̂ are very much related and can be referred to as the dual of L and are related thus, L̂ (t, r, s1, s2, z) = L (t, r, s1, s2, g) − zg. (21) where g (t, r, s1, s2, z) = x, Lx = z, g = −L̂z. (22) Differentiating (21) with respect to t, r, s1, s2 and x Lt = L̂t, Ls1 = L̂s1 ,Ls2 = L̂s2, Lr = L̂r, , Lrx = −L̂rz L̂zz , Ls1 r = L̂s1 r − L̂s1 zL̂rz L̂zz , Ls2 r = L̂s2 r − L̂s2 zL̂rz L̂zz , Lxx = −1 L̂zz , Ls1 s1 = L̂s1 s1 − L̂2s1 z L̂zz ,Ls2 s2 = L̂s2 s2 − L̂2s2 z L̂zz , Lx = z, Ls1 x = −L̂s1 z L̂zz ,Ls2 x = −L̂s2 z L̂zz Lrr = L̂rr − L̂2rz L̂zz (23) At terminal time T , we define the dual utility in terms of the original utility function U (x) as Û (z) = sup U (x) − zx | 0 < x < ∞} , and G (z) = sup?{x|U(x) ≥ zx + Û (z) . As a result L̂ (t, r, s1, s2, z) = L (t, r, s1, s2, g) − zg. G (z) = (U′)−1 (z) , (24) where G is the inverse of the marginal utility U and note that L (T, r, s1, s2, x) = U(x). At terminal time T , we can define g (T, r, s1, s2, z) = in f x>0 x ∣∣∣∣ U (x) ≥ zx + L̂ (t, r, s1, s2, z)} and L̂ (t, r, s1, s2, z) = sup x>0 { U (x) − zx} so that g (T, r, s1, s2, z) = (U ′)−1 (z) . (25) Substituting (23) into (11), (12) and (17), we have L̂t + µ1 s1L̂s1 + µ2 s2L̂s2 + [r + α1] gz + 12As 2β+2 1 L̂s1 s1 + 1 2Bs 2β+2 2 L̂s2 s2 + 12 δ 2L̂rr + Dδρs β+1 1 L̂s1 r + Eδρsβ+12 L̂s2 r − 1 2 ( α2 s −2β 1 + α3 s −2β 2 ) z2L̂zz − 1 2 δ 2 ( 1 − k3ρ2 ) L̂2zr L̂zz + ( α4 − (µ1 − r) s1 ) zL̂s1 z + (α5 − (µ2 − r) s2) zL̂s2 z +Csβ+11 s β+1 2 L̂s1 s2 + θ (µ0 − r)L̂r + ( α6 s −β 1 + α7 s −β 2 ) zL̂rz  = 0.(26) ϕ∗1 =  − [ Csβ1 (µ2−r)−Bs β 2 (µ1−r) ] x(AB−C2)s2β1 s β 2 zL̂zz − s1L̂s1 z x − (BD−CE)δρ x(AB−C2)sβ1 L̂rz  (27) ϕ∗2 =  − [ Csβ2 (µ1−r)−As β 1 (µ2−r) ] x(AB−C2)sβ1 s 2β 2 zL̂zz − s2L̂s2 z x − (AE−CD)δρ x(AB−C2)sβ2 L̂rz  . (28) 242 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 243 From equation (22), differentiating (26), (27) and (28) with respect to z, we have gt + (α4 + rs1) gs1 + (α5 + rs2) gs1 − (r + α1) g − (r + α1)z gz + 1 2As 2β+2 1 gs1 s1 + 12Bs 2β+2 2 gs2 s2 + Cs β+1 1 s β+1 2 gs1 s2 + 1 2 δ 2grr +θ (µ0 − r) gr + Dδρs β+1 1 gs1 r +Eδρsβ+12 gs2 r + 1 2 ( α2 s −2β 1 +α3 s −2β 2 ) z2gzz + ( α2 s −2β 1 +α3 s −2β 2 ) z gz + ( α4 − (µ1 − r) s1 ) zgs1 z + ( α5 − (µ2 − r) s2 ) zgs2 z + ( α6 s −β 1 + α7 s −β 2 ) (gr + zgrz) − 1 2 δ 2 ( 1 − k3ρ2 ) ( 2gr grz gz − gzz g2r g2z )  = 0.(29) ϕ∗1 =  [ Csβ1 (µ2−r)−Bs β 2 (µ1−r) ] x(AB−C2)s2β1 s β 2 z gz + s1 x gs1 + (BD−CE)δρ x(AB−C2)sβ1 gr  (30) ϕ∗2 =  [ Csβ2 (µ1−r)−As β 1 (µ2−r) ] x(AB−C2)sβ1 s 2β 2 z gz + s2 x gs2 + (AE−CD)δρ x(AB−C2)sβ2 gr  (31) where, L (T, r, s1, s2, z) = U(z) and U(z) is the marginal utility of the investor. Next, we proceed to solve (29) for g considering an investor with logarithm utility, then substitute the solution into (30) and (31) for the optimal investment plan using change of variable and asymptotic method. 4. Investor’s Optimal Investment Plan under Logarithm Util- ity Here, we consider an investor with utility function exhibiting constant relative risk aversion (CRRA) different from the one in [18] where the investor exhibited constant absolute risk aver- sion (CARA). Since our interest here is to determine the op- timal investment plan for the investor with CRRA utility, we choose the logarithm utility function similar to the one in [20, 21]. From [1, 21], the logarithm utility function is given as U(x) = lnx, x > 0 From (25), g (T, r, s1, s2, z) = (U ′)−1 (z) = 1 z (32) Next, we conjecture a solution to (21) similar to the one in [21] with the form:{ g (t, r, s1, s2, z) = 1 z [e (t, r, s1) + f (t, r, s2)] + h(t) e (T, r, s1) = 1 2 , f (T, r, s2) = 1 2 , h(T ) = 0, (33)  gt = 1 z [et + f t] + ht, gs1 = 1 z es1 , gs2 = 1 z fs2, gs1 s1 = 1 z es1 s1, grs1 = 1 z [ers1 + f rs1 ] gs2 s2 = 1 z fs2 s2, gz = − 1 z2 [ e + f ] , gzz = 2 z3 [ e + f ] , gs1 z = − 1 z2 es1 , gs2 z = − 1 z2 fs2, grs2 = 1 z [ers2 + f rs2 ], gr = 1 z [er + f r ], grr = 1 z [err + f rr ] , grz = − 1 z2 [er + f r ] (34) Substituting (34) into (29), we have [ht − (r + α1)h] + 1z  et + µ1 s1es1 + 12As2β+21 es1 s1 +Dδρsβ+11 es1 r + θ (µ0 − r) er + 1 2 δ 2err  + 1z  ft + µ2 s2 fs2 + 12Bs2β+22 fs2 s2 +Eδρsβ+12 f s2 r + θ (µ0 − r) fr + 1 2 δ 2 frr   = 0(35) Splitting (35) we have{ ht − (r + α1)h = 0 h(T ) = 0 (36)  et + µ1 s1es1 + 1 2As 2β+2 1 es1 s1 + Dδρs β+1 1 es1 r +θ (µ0 − r) er + 1 2 δ 2err = 0 e (T, r, s1) = 1 2 (37)  ft + µ2 s2 fs2 + 1 2Bs 2β+2 2 fs2 s2 + Eδρs β+1 2 f s2 r +θ (µ0 − r) fr + 1 2 δ 2 frr = 0 f (T, r, s2) = 1 2 (38) Solving equation (36) for h, we obtain h = 0 (39) g (t, r, s1, s2, z) = 1 z (40) To prove the Proposition above, we attempt to solve equations (37) and (38) by stating and proofing the following lemmas The solution of equation (37) is given as e (t, r, s1) = u (t, r, m) = uα (t, r, m) = 1 2 , where uα (t, r, m) = u1 (t, r, m) + √ αu2 (t, r, m) + αu3 (t, r, m) and u1 (t, r, m) = 1 2 , u2 (t, r, m) = 0, and u3 (t, r, m) = 0. Assume{ e (t, r, s1) = u (t, r, m) , m = s −2β 1 e (T, r, s1) = 1 2 , (41) then et = ut, es1 = −2βs −2β−1 1 um , es1 s1 = 2β (2β + 1) s −2β−2 1 um + 4β 2 s−4β−21 umm, er = ur, err = urr, ers1 = −2βs −2β−1 1 urm  (42) 243 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 244 Substituting (42) into (37), we have ut − 2µ1βmum + Aβ (2β + 1) um +2β2Amumm + θ (µ0 − r) ur + 12 δ 2urr − 2βDδρ √ murm  = 0 (43) We can rewrite (43) as (V1 + V2 + V3) u = 0 (44) where V1 = [ θ (µ0 − r) ur + 1 2 δ2urr ] u (45) V2 = [ ut + β ( A (2β + 1) −2µ1m ) um + 2β 2 Amumm ] u (46) V3 = [ −2βDδρ √ murm ] u (47) Next, we follow the approach in [18] by applying the asymp- totic expansion method to solve the problem in (44). Assume that the volatility follows a slow fluctuating process, we attempt to find an asymptotic solution of (44) by a following slow-fluctuating process rαto replace (2), in which 0 < α � 1 is a small positive parameter: drα (t) = θ(µ0 − rα(t))dt −δdB0(t) (48) Substituting (48) into (44) and also replacing µ0−rα(t) by α(µ0− r), we have( αV1 + V2 + √ αV3 ) uα = 0 (49) Next, we conjecture a solution for (49) as follows uα (t, r, m) = ( u1 (t, r, m) + √ αu2 (t, r, m) +αu3 (t, r, m) ) . (50) Substituting (50) into (49), ( αV1 + V2 + √ αV3 )  u1 (t, r, m) + √ αu2 (t, r, m) +αu3 (t, r, m)  = 0 Simplifying the above equation, we arrive at V2u1 (t, r, m) + [ V2u2 (t, r, m) +V3u1 (t, r, m) ] √ α + [ V1u1 (t, r, m) + V2u3 (t, r, m) +V3u2 (t, r, m) ] α  = 0. This implies that{ V2u1 (t, r, m) = 0 u1 (T, r, m) = 1 2 (51) { V2u2 (t, r, m) + V3u1 (t, r, m) = 0 u1 (T, r, m) = 1 2 , u2 (T, r, m) = 0 (52) { V1u1 (t, r, m) + V2u3 (t, r, m) + V3u2 (t, r, m) u1 (T, r, m) = 1 2 , u2 (T, r, m) = 0 u3 (T, r, m) = 0 (53) To obtain the solution of (50), we solve (51), (52) and (53). Recall from (45), (46) and (47), equation (51), (52) and (53) can be expressed as ( u1t + β ( A (2β + 1) −2µ1m ) u1m + 2β2Amu1mm ) = 0 u1 (T, r, m) = 1 2 (54)  u2t + β (A (2β + 1) − 2µ1m) u2m +2β2Amu2mm − 2βDδρ √ mu1rm = 0 u2 (T, r, m) = 0 (55)  u3t + β (A (2β + 1) − 2µ1m) u3m + 2β2Amu3mm +θ (µ0 − r) u1r + 1 2 δ 2u1rr − 2βDδρ √ mu2rm u3 (T, r, m) = 0 (56) Let u1 (t, r, m) = Q0 (t, r) + mQ1 (t, r) (57) and u1t = Q0t + mQ1t, u1m = Q1 , u1mm = 0 (58) Substituting (58) in (54), we have Q0t + βA (2β + 1)Q1 = 0,Q0 (T, r) = 1 2 (59) Q1t − 2µ1mQ1 = 0,Q1 (T, r) = 0 (60) Solving (60), we have Q1 (t, r) = 0 (61) Putting (61) into (59) and solving the resultant equation, we have Q0 (t, r) = 1 2 (62) Hence from (54) u1 (t, r, m) = Q0 (t, r) + mQ1 (t, r) = 1 2 (63) Next, we solve (55), by assuming a solution of the form u2 (t, r, m) = ( Q2 (t, r) + m 1 2 Q3 (t, r) + mQ4 (t, r) +m 3 2 Q5 (t, r) ) (64) and  u2t = Q2t + m 1 2 Q3t + mQ4t + m 3 2 Q5t , u2m = 1 2 m − 1 2 Q3 + Q4 + 3 2 m 1 2 Q5 u2mm = − 1 4 m − 3 2 Q3 + 3 4 m − 1 2 Q5, u1rm = 0  (65) Substituting (65) into (55), we have{ Q2t + β (2β + 1)AQ4 = 0, Q2 (T, r) = 0 (66) { Q3t −µ1βQ3 + 3 2 β (3β + 1)AQ5 = 0, Q3 (T, r) = 0 (67) { Q4t − 2µ1βQ4 = 0, Q4 (T, r) = 0 (68) 244 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 245{ Q5t − 3µ1βQ5 = 0, Q5 (T, r) = 0 (69) Solving (66), (67), (68) and (69) with their boundary condi- tions, we have Q2 (t, r) = Q3 (t, r) = Q4 (t, r) = Q5 (t, r) = 0, Hence from (64) u2 (t, r, m) = ( Q2 (t, r) + m 1 2 Q3 (t, r) +mQ4 (t, r) + m 3 2 Q5 (t, r) ) = 0 (70) Next, we attempt to solve (56), by assuming a solution of the form u3 (t, r, m) = ( Q6 (t, r) + m 1 2 Q7 (t, r) + mQ8 (t, r) +m 3 2 Q9 (t, r) + m2Q9 (t, r) ) (71) u3t = Q6t + m 1 2 Q7t + mQ8t + m 3 2 Q9t+m2Q10t , u3m = 1 2 m − 1 2 Q7 + Q8 + 3 2 m 1 2 Q9+2mQ10 u2mm = − 1 4 m − 3 2 Q7 + 3 4 m − 1 2 Q9 + 2Q10, u1r = 0, u1rr = 0, u2rm = 0  (72) Substituting (72) in (56), we have Q6t + β (2β + 1)AQ8 = 0,Q6 (T, r) = 0 (73) Q7t −µ1βQ7 + 3 2 βA (3β + 1)Q9 = 0,Q7 (T, r) = 0 (74) Q8t − 2µ1βQ8 + 2βA (4β + 1)Q10 = 0,Q8 (T, r) = 0 (75) Q9t − 3µ1βQ9 = 0,Q9 (T, r) = 0 (76) Q10t − 4µ1βQ10 = 0,Q10 (T, r) = 0 (77) Solving (73), (74), (75), (4,45) and (77), we have Q6 (t, r) = Q7 (t, r) = Q8 (t, r) = Q9 (t, r) = Q10 (t, r) = 0. Therefore, (71) reduces to u3 (t, r, m) = Q6 (t, r) + m 1 2 Q7 (t, r) + mQ8 (t, r) + m 3 2 Q9 (t, r) + m 2 Q9 (t, r) = 0 (78) Hence, from (48), (63), (70) and (78), we have e (T, r, s1) = uα (t, r, m) = u1 (t, r, m) + √ αu2 (t, r, m) + αu3 (t, r, m) = 1 2 The solution of equation (38) is given as f (t, r, s2) = v (t, r, `) = vα (t, r, `) = 1 2 where vα (t, r, `) = v1 (t, r, `) + √ αv2 (t, r, `) + αv3 (t, r, `) and v1 (t, r, `) = 1 2 , v2 (t, r, `) = 0, and v3 (t, r, `) = 0 Assume{ f (t, r, s2) = v (t, r, `) , ` = s −2β 2 f (t, r, s2) = 1 2 (79) Then ft = vt, fs2 = −2βs −2β−1 2 v` , fs2 s2 = 2β (2β + 1) s −2β−2 2 v` + 4β 2 s−4β−22 v``, fr = vr, frr = vrr, frs2 = −2βs −2β−1 2 vr`  (80) Substituting (80) into (38), we have[ vt − 2µ2β`v` + Bβ (2β + 1) v` + 2β2B`v`` +θ (µ0 − r) vr + 1 2 δ 2vrr − 2βEδρ √ `vr` ] = 0 (81) We can rewrite (81) as (M1 + M2 + M3) v = 0 (82) Where M1 = [ θ (µ0 − r) vr + 1 2 δ2vrr ] v (83) M2 = [ vt + β (B (2β + 1) − 2µ2`v`) v` +2β2B`v`` ] v (84) M3 = [ −2βEδρ √ `vr` ] v (85) Substituting (48) into (82) and replacing µ0 −rα(t) by α(µ0 −r), we have,( αM1 + M2 + √ αM3 ) vα = 0 (86) Next, we conjecture a solution for (86) as follows vα (t, r, `) = v1 (t, r, `) + √ αv2 (t, r, `) + αv3 (t, r, `) (87) Substituting (87) into (86), we have( αM1 + M2 + √ αM3 ) (v1 (t, r, `) + √ αv2 (t, r, `) + αv3 (t, r, `) ) = 0. Simplifying the above equation, we arrive at( M2v1 (t, r, `) + [M2v2 (t, r, `) + M3v1 (t, r, `)] √ α + [M1v1 (t, r, `) + M2v3 (t, r, `) + M3v2 (t, r, `)] α ) = 0. This implies that{ M2v1 (t, r, `) = 0 v1 (T, r, `) = 1 2 , (88) { M2v2 (t, r, `) + M3v1 (t, r, `) = 0 v1 (T, r, `) = 1 2 , v2 (T, r, `) = 0 , (89) { M1v1 (t, r, `) + M2v3 (t, r, `) + M3v2 (t, r, `) v1 (T, r, `) = 1 2 , v2 (T, r, `) = 0 v3 (T, r, `) = 0 .(90) To obtain the solution of (87), we solve (88), (89) and (90). Recall from (83), (84) and (85), equation (88), (89) and (90) can be expressed as{ v1t + β (B (2β + 1) − 2µ1`) v1` + 2β2B`v1`` = 0 v1 (T, r, `) = 1 2 , (91) 245 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 246 v2t + β (B (2β + 1) − 2µ2`) v2` +2β2B`v2`` − 2βEδρ √ `v1r` = 0 v2 (T, r, `) = 0 , (92)  v3t + β (B (2β + 1) − 2µ2`) v3` + 2β2B`v3`` +θ (µ0 − r) v1r + 1 2 δ 2v1rr − 2βEδρ √ `v2r` v3 (T, r, `) = 0 . (93) Let v1 (t, r, `) = R0 (t, r) + `R1 (t, r) (94) and v1t = Q0t + `R1t, u1` = R1 , u1`` = 0. (95) Substituting (95) in (91), we have R0t + βB (2β + 1)R1 = 0,R0 (T, r) = 1 2 (96) R1t − 2µ1`R1 = 0,R1 (T, r) = 0 (97) Solving (97), we have R1 (t, r) = 0. (98) Putting (98) into (96) and solving the resultant equation, we have R0 (t, r) = 1 2 . (99) Hence from (94) v1 (t, r, `) = R0 (t, r) + `R1 (t, r) = 1 2 (100) Next, we solve (92), by assuming a solution of the form v2 (t, r, `) = ( R2 (t, r) + ` 1 2 R3 (t, r) +`R4 (t, r) + ` 3 2 R5 (t, r) ) (101) and v2t = R2t + ` 1 2 R3t + `R4t + ` 3 2 R5t , v2` = 1 2 ` − 1 2 R3 + R4 + 3 2 ` 1 2 R5 v2`` = − 1 4 ` − 3 2 R3 + 3 4 ` − 1 2 R5, u1r` = 0  (102) Substituting (102) into (92), we have R2t + β (2β + 1)BR4 = 0,R2 (T, r) = 0 (103) R4t − 2µ2βR4 = 0,R4 (T, r) = 0 (104) R5t − 3µ2βR5 = 0,R5 (T, r) = 0 (105) R3t −µ2βR3 + 3 2 β (3β + 1)BR5 = 0,R3 (T, r) = 0 (106) Solving (103), (104), (105) and (106) with their boundary con- ditions, we have R2 (t, r) = R3 (t, r) = R4 (t, r) = R5 (t, r) = 0, Hence from (101) v2 (t, r, `) = ( R2 (t, r) + ` 1 2 R3 (t, r) + `R4 (t, r) +` 3 2 R5 (t, r) ) = 0.(107) Next, we attempt to solve (93), by assuming a solution of the form v3 (t, r, `) = R6 (t, r)+` 1 2 R7 (t, r)+`R8 (t, r)+` 3 2 R9 (t, r)+` 2 R9 (t, r)(108) v3t = R6t + ` 1 2 R7t + `R8t + ` 3 2 R9t+` 2R10t , u3` = 1 2 ` − 1 2 R7 + R8 + 3 2 ` 1 2 R9+2`R10 v2`` = − 1 4 ` − 3 2 R7 + 3 4 ` − 1 2 R9 + 2R10, v1r = 0, v1rr = 0, v2r` = 0  (109) Substituting (109) in (93), we have R6t + β (2β + 1)BR8 = 0,R6 (T, r) = 0, (110) R7t −µ2βR7 + 3 2 βB (3β + 1)R9 = 0,R7 (T, r) = 0, (111) R8t −2µ2βR8 + 2βB (4β + 1)R10 = 0,R8 (T, r) = 0,(112) R9t − 3µ2βR9 = 0,R9 (T, r) = 0, (113) R10t − 4µ2βR10 = 0,R10 (T, r) = 0. (114) Solving (110), (111), (112), (4,82) and (114), we have R6 (t, r) = R7 (t, r) = R8 (t, r) = R9 (t, r) = R10 (t, r) = 0. Therefore (108) reduces to v3 (t, r, `) = R6 (t, r) + ` 1 2 R7 (t, r) + `R8 (t, r) (115) + ` 3 2 R9 (t, r) + ` 2 R9 (t, r) = 0. Finally, substituting (100), (107) and (115) into (87), we have f (t, r, s2) = vα (t, r, `) (116) = v1 (t, r, `) + √ αv2 (t, r, `) + αv3 (t, r, `) = 1 2 . From Lemma 4, 4 and equation (39), Proposition 4 is proved. The optimal investment plans are given as ϕ∗1 = 1 x2 sβ2 s 2β 1  (( σ221 + σ 2 22 ) (µ1 − r) s β 2 − (σ11σ21 + σ12σ22) (µ2 − r) s β 1 ) (σ211 + σ 2 12)(σ 2 21 + σ 2 22) − (σ11σ21 + σ12σ22) 2  ϕ∗2 = 1 x2 sβ1 s 2β 2  ( (σ211 + σ 2 12) (µ2 − r) s β 1 − (σ11σ21 + σ12σ22) (µ1 − r) s β 2 ) (σ211 + σ 2 12)(σ 2 21 + σ 2 22) − (σ11σ21 + σ12σ22) 2  ϕ∗0 = 1 −ϕ ∗ 1 −ϕ ∗ 2 Recall that equation (22) and (23) are given as ϕ∗1 = [ Csβ1 (µ2 − r) −Bs β 2 (µ1 − r) ] x ( AB−C2 ) s2β1 s β 2 z gz + s1 x gs1 + (BD−CE) δρ x ( AB−C2 ) sβ1 gr ϕ∗2 = [ Csβ2 (µ1 − r) −As β 1 (µ2 − r) ] x ( AB−C2 ) sβ1 s 2β 2 z gz + s2 x gs2 + (AE−CD) δρ x ( AB−C2 ) sβ2 gr 246 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 247 Figure 1. Time evolution of the optimal investment plan ϕ∗0, ϕ ∗ 1, and ϕ ∗ 2. From Proposition 4 and equation (22) we have g (t, r, s1, s2, z) = 1 z and g = x. Differentiating g with respect to r, s1, s2, z and substitute into equation (30) and (31), we have ϕ∗1 = 1 x2 sβ2 s 2β 1   ( σ221 + σ 2 22 ) (µ1 − r) s β 2 − (σ11σ21 + σ12σ22) (µ2 − r) s β 1  (σ211 + σ 2 12)(σ 2 21 + σ 2 22) − (σ11σ21 + σ12σ22) 2  ϕ∗2 = 1 x2 sβ1 s 2β 2   ( σ211 + σ 2 12 ) (µ2 − r) s β 1 −(σ11σ21 + σ12σ22) (µ1 − r) s β 2 ( σ211 + σ 2 12 ) ( σ221 + σ 2 22 ) −(σ11σ21 + σ12σ22) 2  ϕ∗0 = 1 −ϕ ∗ 1 −ϕ ∗ 2 i. recall that from equation (33), J (t, r, s, z) = w (t, r, s) + v(t, r, s)lnz From Proposition 4 and 4, the proof is completed. From Lemma 4, we observed that our result is similar to the one in [3] but the difference between our result and theirs is that their interest rate is a constant while ours is stochastic. 5. Numerical Simulations In this section, some numerical simulations are presented to study the effect of some sensitive parameters on the optimal investment plans under logarithm utility. To achieve this, the following data will be used unless otherwise stated; Table 1. Parameters σ11 σ12 σ21 σ22 µ1 Values 1.0 0.9 0.85 0.80 0.25 µ2 x r (0) S1 (0) S2 (0) β T 0.2 1 0.1 1.5 1.2 −1 3 Figure 2. Time evolution of ϕ∗0 with different elasticity parameter β. Figure 3. Time evolution of ϕ∗1 with different elasticity parameter β. Figure 4. Time evolution of ϕ∗2 with different elasticity parameter β. 6. Discussion In this section, the effects of some parameters on the optimal investment plans are examined. In Figure 1, the simulation of optimal investment plans for the three assets are given against time; the graph shows that at the initial time, the investor will invest more in the risk free asset and suddenly reduce it and then increase it continually till the expiration of investment. Also, the investor will invest less in the two risky assets and suddenly 247 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 3 (2021) 239–249 248 Figure 5. The effect of the risk free interest r on ϕ∗0. Figure 6. The effect of the risk free interest r on ϕ∗0. increase it and then reduced it continually till the expiration of investment. The behavior of the optimal investment plans in Figure 1 is due to O-U process used in modeling the risk free interest rate which reflects a fluctuation in interest rate. From this result, we can see that O-U process enable the investor to know when there is a change in interest rate for the risk free asset thereby adjusting the optimal investment plan of the dif- ferent assets to suit the market condition at each point in time. The implication of this is that each time the risk free interest rate appreciate significantly, the investor will likely to invest more in the risk free asset while reducing that of the two risky asset in a way that he or she will maximize their portfolio. Fur- thermore, we observed that the investor prefers investing more in stock 1 than stock 2; this is because the appreciation rate of stock 1 is higher than the appreciation rate of stock 2, thereby making stock 1 more attractive than stock 2 all other factors re- maining constant. Figure 2, presents simulation of ϕ∗0 against time with different values of the elasticity parameter β. The graph shows that as β reduces, the optimal investment plan ϕ∗0 for the risk free asset increases. Conversely, Figures 3 and 4, present simulations of the optimal investment plan against time with different values of the elasticity parameter β; the graphs show that as β reduces, the optimal investment plans for the two risky assets decrease. This is because a more negative elas- ticity parameter β gives rise to an increased probability of a large adverse movement in the stock market prices which lead to an increased expected decrease in volatility that comes with a lower β. Therefore, the investor is advised to reduce the fraction of his or her investment in the two risky assets thereby hedging the volatility risk that comes with a decreased value of β. Also, we observed that from equations (3) and (4), the stocks market prices are increasing functions of the appreciation rate µ1 and µ2 and a decreasing function of the instantaneous volatility as the elasticity parameter decreases. This implies that as β de- creases, the risky assets become less attractive thereby leading to decrease in the optimal investment plan of the two risky as- sets. Figure 5 and 6 give the analysis of the effect of the risk free interest rate on the optimal investment plan; in Figure 5, the optimal investment plan for the risk free asset increases as the interest rate r increases. It is observed that the proportion in- vested in risk free assets increases faster at early of investment and very slow as retirement age gets closer; this implies that at the early stage of investment, the investor may not be inter- ested in taking more risk but as retirement age draw closer, he invest more in the two risky assets with the aim of maximizing his portfolio. On the contrary, Figure 6 shows that the optimal investment plan for the risky assets decreases with increase in risk free interest rate r. This can be attributed to two factors; first, Investors are naturally attracted to high interest rate and if this happens, they will love to invest more in risk free assets for higher returns thereby reducing investment in risky assets. Secondly, considering (µ1 − r) in ϕ∗1 where µ1, is the appreci- ation rate of stock 1 and r is the risk free interest rate. We observed that as r increases, (µ1 − r) decreases. Hence, from Figure 6, we observed that as the interest rate increases more than the appreciation rate, the investor invests less in stock 1 and when the interest rate is less than the appreciation rate, the investor invests more in stock 1. So, in general, we observed that all other factors being constant, the investor will choose his portfolio depending on the value of the risk free interest rate and the corresponding appreciation rates of the risky assets. Finally, from Proposition 4, we observed that optimal investment plan for an investor with logarithm utility is not affected by the risk averse coefficient unlike the result in [18] where the optimal in- vestment plan depends on the investor’s behaviour towards risk. 7. Conclusion In conclusion, this work studied the optimal investment plan for an investor with logarithm utility under the constant elas- ticity of variance model in the presence of stochastic interest rate modelled by the Ornstein-Uhlenbeck process. We devel- oped a portfolio consisting of three assets (risk free asset and two risky assets). The Legendre transformation and dual theory with asymptotic expansion technique was used to find closed form solutions of the optimal investment plans. Also studied were the effects of some parameters on the optimal investment plans using numerical simulations. We observed that when the risk free asset is modelled by the O-U process and the two risky asset are modelled by the CEV process, the optimal investment plan for the three assets give a fluctuation effect, showing that 248 Akpanibah & Ini / J. Nig. Soc. Phys. 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