# Robust Time-Consistent Portfolio Selection for an Investor under CEV Model with Inflation Influence

Authors: Peng Yang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A robust time-consistent optimal investment strategy selection problem under inflation influence is investigated in this article. The investor may invest his wealth in a financial market, with the aim of increasing wealth. The financial market includes one risk-free asset, one risky asset, and one inflation-indexed bond. The price process of the risky asset is governed by a constant elasticity of variance (CEV) model. The investor is ambiguity-averse; he doubts about the model setting under the original probability measure. To dispel this concern, he seeks a set of alternative probability measures, which are absolutely continuous to the original probability measure. The objective of the investor is to seek a time-consistent strategy so as to maximize his expected terminal wealth meanwhile minimizing his variance of the terminal wealth in the worst-case scenario. By using the stochastic optimal control technique, we derive closed-form solutions for the optimal time-consistent investment strategy, the probability scenario, and the value function. Finally, the influences of model parameters on the optimal investment strategy and utility loss function are examined through numerical experiments.


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## Body

## 1. Introduction

Nowadays, portfolio selection is a very important research topic in mathematical finance. As we all know, Markowitz [1] pioneered this research topic. He measured the investment gain and risk by expectation and variance, respectively. Nowadays, scholars call this method as the mean-variance (MV) criterion. Li and Ng [2] and Zhou and Li [3], respectively, pioneered the multiperiod and the continuous-time MV problem, where the explicit solutions were obtained. Dai et al. [4] studied the continuous-time MV problem with transaction costs. Recently, Sun et al. [5, 6] studied the MV problem for an insurer.The aforementioned work under the MV criterion is time inconsistency. That is to say, the optimal strategy made at timet may not be optimal at time s, s>t. However, time consistency of strategy is an important and a rational requirement in many practice scenarios. For example, a few years ago, the Chinese government decided to get rid of poverty by 2020. Naturally, they hope to get rid of poverty at 2020. To the best of our knowledge, Strotz [7] first formally treated time inconsistency. He proposed a game theoretic approach and sought an equilibrium strategy. Thus, the equilibrium strategy is time-consistent. Nowadays, this method becomes a mainstream way to deal with time-inconsistency. Ekeland et al. [8] first proposed the rigorous definition of the equilibrium strategy in a continuous-time framework. Bjӧrk and Murgoci [9] studied the time-consistent investment strategy, where the model parameters are controlled by a common Markov process. Li et al. [10] considered the optimal time-consistent MV problem under the Heston model. Yang [11] studied the similar problem, where the aggregate claim and the price are shocked by a common Poisson process. Zeng et al. [12] studied the robust time-consistent strategy under the MV framework. Furthermore, Yang et al. [13] extended the time-consistent MV problem to a new interaction mechanism. For more other detailed and related studies, one can refer to Bjӧrk et al. [14], Czichowsky [15], and Kronborg and Steffensen [16].However, most of the aforementioned papers ignore the inflation risk and assume that the risky asset follows a geometric Brownian motion, which is usually contrary to practice investment activities according to many empirical studies. Brennan and Xia [17] studied the asset distribution problem under inflation influence. Han and Huang [18] considered the defined-contribution (DC) pension problem under inflation influence. Kwak and Lim [19] studied the optimal consumption-investment problem under inflation influence. Li et al. [20] studied the time-consistent investment strategy selection problem for a DC pension under partial information and inflation influence.On the contrary, the CEV model has become popular among academia and practitioners since proposed by Cox and Ross [21]. Gao [22] investigated the DC pension investment problem under the CEV model. Gu et al. [23], Lin and Li [24], and Liang et al. [25] investigated the optimal reinsurance and investment strategy selection problem under the CEV model. Lin and Qian [26] studied the similar problem, where the strategy is time-consistent. For more other detailed and related studies about the CEV model, readers can refer to Li et al. [27] and Zheng et al. [28].Bearing in mind the aforementioned state of the art, in this article, we will investigate the time-consistent investment strategy under the CEV model with inflation influence. The financial market includes one risk-free asset, one risky asset, and one inflation-indexed bond. The price process of the risky asset is governed by the CEV model. We assume that the investor is ambiguity-averse and worries about uncertainty in model setting. To dispel this concern, he seeks a set of alternative probability measures, which are absolutely continuous to the original probability measure. Then, the new model setting is obtained under the new probability measure. The objective of the investor is to find a time-consistent strategy so as to maximize his expected terminal wealth meanwhile minimizing his variance of the terminal wealth in the worst-case scenario. The problem’s solving steps are as follows: first, we provide an extended HJB equation and a verification theorem. Second, by using the stochastic optimal control technique, we derive analytically the optimal time-consistent investment strategy, the probability scenario, and the value function. Finally, the influences of model parameters on the optimal investment strategy and utility loss function are examined through numerical experiments.Compared with some related current research studies, our main contributions are given as follows:(i)
We consider a general financial market, which includes one risk-free asset, one risky asset, and one inflation-indexed bond. The price process of the risky asset is governed by the CEV model.(ii)
We study the time-consistent investment problem under inflation influence.(iii)
Closed-form solutions for the optimal time-consistent robust investment strategy, optimal probability scenario, and value function are derived.(iv)
The influences of model parameters on the optimal time-consistent investment strategy and utility loss function are systematically examined through numerical experiments.The remainder of this article is arranged as follows. In Section2, we describe the model setup. In Section 3, an optimal robust time-consistent investment strategy selection problem is formulated. In Section 4, the closed-form solutions for the time-consistent optimal investment strategy and optimal probability scenario are derived. In Section 5, the influences of model parameters on the optimal time-consistent investment strategy and utility loss function are examined through numerical experiments. The final section summarizes this article.


## 2. Model Setup

We construct a financial model and present some basic assumptions in this section. All stochastic processes and random variables, mentioned later, are defined on a filtered complete probability spaceΩ,ℱ,ℱt,P. We assume that ℱ≔ℱt,t≥0 is right-continuous and is complete with respect to P. ℱt stands for the information acquired by the investor up to time t. We assume that there are no market frictions in trading.


### 2.1. Financial Market

A financial market with the inflation risk is given in this section. To the best of our knowledge, consumer price index (CPI) is often used to describe the inflation rate in economics and academic research. CPI can be seen as a price level process. In this article, the inflation price levelP0t satisfies the following process:(1)dP0t=P0tμ1dt+σ1dW1t,where μ1>0 represents the instantaneous expected rate of the inflation, σ1>0 stands for the volatility of the inflation, and W1t stands for a standard Brownian motion.We assume that there are three assets available for the investor: one risk-free asset, one risky asset, and one inflation-indexed bond. The price processP1t of the risk-free asset is given by(2)dP1t=r1P1tdt.Here,r1>0 represents the interest rate of the risk-free asset. The price process St of the risky asset satisfies the following CEV model:(3)dSt=Stμ2dt+σ2SβtdW2t,S0=s0.Here,μ2≥r1 stands for the appreciation rate, σ2>0 represents the price volatility, W2t represents a standard Brownian motion, and β is the constant elasticity parameter. We assume W1t and W2t are mutually independent.Remark 1.
The parameterβ can take all real numbers. Obviously, if β=0, the CEV model will degenerate to a geometric Brownian motion. When the CEV model is proposed, β is assumed to be negative. Emanuel and Macbeth [29] proved that β can also be assumed to be positive. The CEV model has no jump, i.e., we do not consider the jump risk in this article. We can also consider the price process with the jump. Then, the exponential Lévy model is a good choice. For the exponential Lévy model, readers can refer to Yu et al. [30] and Zhang et al. [31].
The third asset to invest in is the inflation-indexed bond. Similar to Kwak and Lim [19], the price Pt of the inflation-indexed bond is given by(4)dPtPt=r2dt+dP0tP0t=r2+μ1dt+σ1dW1t.
Here,r2>0 represents the real interest rate, and r2+μ1>r1 stands for the appreciation rate.


### 2.2. Inflation-Adjusted Wealth Process

Letu1t and u2t, respectively, denote the proportions of the wealth invested in the risky asset and the inflation-indexed bond at time t, and the remainder of the proportion is invested in the risk-free asset. At time t, the investor (here, we only consider an investor. To reflect the general situation, one can consider n investors. For example, Yang et al. [13], Espinosa and Touzi [32], and Yu et al. [33] studied the optimal control problem for n agents) can choose the proportions u1t and u2t as control strategies; we denote them as ut=u1t,u2t. For each strategy ut, the wealth process Rtu can be described as(5)dRtuRtu=1−u1t−u2tdP1tP1t+u1tdStSt+u2tdPtPt=r1+u1tμ2−r1+u2tr2+μ1−r1dt+u1tσ2SβtdW2t+u2tσ1dW1t.Denote(6)Xtu=RtuP0t.By applying Itô’s formula,Xtu can be described as(7)dXtu=1P0tdRtu+Rtud1P0t+dRtu,1P0t=Xtur1+σ12−μ1+u1tμ2−r1+u2tr2+μ1−r1−σ12dt+u1tσ2SβtdW2t+u2t−1σ1tdW1t.Here,Xtu is the real wealth with stripping out inflation.To facilitate solving the optimization problem in Section3, we define the following notations:(8)π1t=u1tXtu,π2t=u2t−1Xtu,πt=π1t,π2t,r˜1=r1+σ12−μ1,r˜2=r2+μ1−r1−σ12,r˜=r˜1+r˜2.Then, the real wealthXtπ can be described as(9)dXtπ=r˜Xtπ+π1tμ2−r1+π2tr˜2dt+π1tσ2SβtdW2t+π2tσ1dW1t.


## 2.1. Financial Market

A financial market with the inflation risk is given in this section. To the best of our knowledge, consumer price index (CPI) is often used to describe the inflation rate in economics and academic research. CPI can be seen as a price level process. In this article, the inflation price levelP0t satisfies the following process:(1)dP0t=P0tμ1dt+σ1dW1t,where μ1>0 represents the instantaneous expected rate of the inflation, σ1>0 stands for the volatility of the inflation, and W1t stands for a standard Brownian motion.We assume that there are three assets available for the investor: one risk-free asset, one risky asset, and one inflation-indexed bond. The price processP1t of the risk-free asset is given by(2)dP1t=r1P1tdt.Here,r1>0 represents the interest rate of the risk-free asset. The price process St of the risky asset satisfies the following CEV model:(3)dSt=Stμ2dt+σ2SβtdW2t,S0=s0.Here,μ2≥r1 stands for the appreciation rate, σ2>0 represents the price volatility, W2t represents a standard Brownian motion, and β is the constant elasticity parameter. We assume W1t and W2t are mutually independent.Remark 1.
The parameterβ can take all real numbers. Obviously, if β=0, the CEV model will degenerate to a geometric Brownian motion. When the CEV model is proposed, β is assumed to be negative. Emanuel and Macbeth [29] proved that β can also be assumed to be positive. The CEV model has no jump, i.e., we do not consider the jump risk in this article. We can also consider the price process with the jump. Then, the exponential Lévy model is a good choice. For the exponential Lévy model, readers can refer to Yu et al. [30] and Zhang et al. [31].
The third asset to invest in is the inflation-indexed bond. Similar to Kwak and Lim [19], the price Pt of the inflation-indexed bond is given by(4)dPtPt=r2dt+dP0tP0t=r2+μ1dt+σ1dW1t.
Here,r2>0 represents the real interest rate, and r2+μ1>r1 stands for the appreciation rate.


## 2.2. Inflation-Adjusted Wealth Process

Letu1t and u2t, respectively, denote the proportions of the wealth invested in the risky asset and the inflation-indexed bond at time t, and the remainder of the proportion is invested in the risk-free asset. At time t, the investor (here, we only consider an investor. To reflect the general situation, one can consider n investors. For example, Yang et al. [13], Espinosa and Touzi [32], and Yu et al. [33] studied the optimal control problem for n agents) can choose the proportions u1t and u2t as control strategies; we denote them as ut=u1t,u2t. For each strategy ut, the wealth process Rtu can be described as(5)dRtuRtu=1−u1t−u2tdP1tP1t+u1tdStSt+u2tdPtPt=r1+u1tμ2−r1+u2tr2+μ1−r1dt+u1tσ2SβtdW2t+u2tσ1dW1t.Denote(6)Xtu=RtuP0t.By applying Itô’s formula,Xtu can be described as(7)dXtu=1P0tdRtu+Rtud1P0t+dRtu,1P0t=Xtur1+σ12−μ1+u1tμ2−r1+u2tr2+μ1−r1−σ12dt+u1tσ2SβtdW2t+u2t−1σ1tdW1t.Here,Xtu is the real wealth with stripping out inflation.To facilitate solving the optimization problem in Section3, we define the following notations:(8)π1t=u1tXtu,π2t=u2t−1Xtu,πt=π1t,π2t,r˜1=r1+σ12−μ1,r˜2=r2+μ1−r1−σ12,r˜=r˜1+r˜2.Then, the real wealthXtπ can be described as(9)dXtπ=r˜Xtπ+π1tμ2−r1+π2tr˜2dt+π1tσ2SβtdW2t+π2tσ1dW1t.


## 3. Problem Formulation

MV investment strategy selection problem as a classic optimization problem can be given as follows:(10)supπE0,x0,s0XTπ,−Var0,x0,s0XTπ.Here,E0,x0,s0⋅=EX0π=x0,S0=s0, Var0,x0,s0⋅=VarX0π=x0,S0=s0, x0 is the investor’s initial wealth, and T is the termination time of the investment. Obviously, the objective of the investor is to find an investment strategy so as to maximize his expected terminal wealth meanwhile minimizing his variance of the terminal wealth. It is well known that this problem is equivalent to the following problem:(11)supπE0,x0,s0XTπ−γ2Var0,x0,s0XTπ,where γ>0 represents the risk-aversion parameter. According to Björk and Murgoci [9] and Kronborg and Steffensen [16], we know that the optimal strategy of (11) is a precommitment strategy, which is time-inconsistent.As we explained in the introduction, one usually takes into account the time-consistent strategy. Similar to Yang [11], Lin and Qian [26], and other related papers, we consider the following problem:(12)supπEt,x,sXTπ−γ2Vart,x,sXTπ,where Et,x,s⋅=EXtπ=x,St=s and Vart,x,s⋅=VarXtπ=x,St=s. The target is to develop the corresponding equilibrium investment strategy, which is time-consistent.Problem (12) is the traditional MV problem, where the investor is ambiguity-neutral. That is, he fully believes in the model defined under the probability measure P. In the actual economic activities, the investor often is ambiguity-averse and wants to protect himself against worst-case scenarios. Similar to Zeng et al. [12], Maenhout [34, 35], Chen and Yang [36], and other papers, we incorporate ambiguity aversion into MV problem (12). Since the investor is ambiguity-averse, he may doubt the model defined under the probability measure P. Hence, we defined an alternative probability measure Q, which is absolutely continuous to P. All such Q are denoted by Q, that is,(13)Q≔QQ∼P.Now, we define an admissible strategy.Definition 1.
For any fixedt∈0,T, an investment strategy πt=π1t,π2t is called an admissible strategy for the ambiguity-averse investor (AAI) if it satisfies the following:(i)
π1t and π2t are progressively measurable with respect to ℱt(ii)
Et,x,sQ∗∫0Tπ12tS2βdt<∞ and Et,x,sQ∗∫0Tπ22tdt<∞, where Q∗ is the probability measure under the worst-case scenario and Et,x,sQ∗⋅=Et,x,sQ∗⋅∣Xtπ=x,St=s(iii)
Equation (9) with respect to πt has a unique strong solution
We denote all such admissible investment strategies on the time interval0,T as Π.
To transform model (9) from the probability measure P to the probability measure Q, we define a process θt=θ1t,θ2tt∈0,T, which satisfies the following:(i)
θ1t and θ2t are progressively measurable with respect to ℱt(ii)
Eexp1/2∫0Tθ12t+θ22tdt<∞
All suchθt on the time interval 0,T are denoted by Θ. For any θ∈Θ, we define a real-valued process Zθtt∈0,T on Ω,ℱ,ℱt,P as(14)Zθt≔exp∫0tθ1sdW1s+∫0tθ2sdW2s−12∫0tθ12sds−12∫0tθ22sds.
Then,Zθt is a martingale with respect to P. In the following, we define a new probability measure Q, which is given by(15)dQdPℱT≔ZθT.
From the definition ofQ, it is clear that Q is absolutely continuous to P.
For anyθ∈Θ, we, respectively, define two new processes W1Qt and W2Qt by(16)dW1Qt=dW1t−θ1tdt,dW2Qt=dW2t−θ2tdt.
According to Girsanov’s theorem,W1Qt and W2Qt are standard Brownian motions with respect to Q. Furthermore, the wealth process (9) under Q can be rewritten as(17)dXtπ=r˜Xtπ+π1tμ2−r1+π2tr˜2+π1tθ2tσ2Sβ+π2tθ1tσ1dt+π1tσ2SβdW2Qt+π2tσ1dW1Qt,and corresponding CEV model (3) becomes(18)dSt=Stμ2+θ2tσ2Sβtdt+σ2SβtdW2Qt.
In the following, we modify MV problem (12) under the worst-case scenario. Through modifying MV problem (12), we shall deal with a robust time-consistent MV strategy selection problem as follows:(19)supπ∈ΠVπt,x,s=supπ∈ΠinfQ∈QV^π,Qt,x,s.
Here,(20)V^π,Qt,x,s=Et,x,sQ∫tTθ1v22ϕ1v+θ2v22ϕ2vdv+Et,x,sQXTπ−γ2Vart,x,sQXTπ,(21)Vπt,x,s=infQ∈QV^π,Qt,x,s,where ϕ1t and ϕ2t are nonnegative and stand for the investor’s ambiguity aversion with respect to the model under P. The larger ϕ1t and ϕ2t are, the more the model under Q will deviate from the model under P. Therefore, AAI’s ambiguity aversion is an increasing function of ϕ1t and ϕ2t. To embody some good properties of the model under P, AAI deviation from P is penalized by the first two terms in (20). The penalty terms depend on the relative entropy. The increase in relative entropy from t to t+dt equals(22)12θ1t2+θ2t2dt.
The proof of (22) is similar to that in Appendix A in Zeng et al. [12]; we omit it here.
To obtain the time-consistent investment strategy, we present the definition of the equilibrium strategy.Definition 2.
For∀t,x,s∈0,T×R×R+, we choose an admissible strategy π∗t,x,s∈Π. Then, through choosing three real numbers a˜∈R+, b˜∈R, and ς>0, we define a new strategy by(23)πςl,x˜,s˜=a˜,b˜forl,x˜,s∈t,t+ς×R×R+,π∗l,x˜,s,forl,x˜,s∈t+ς,T×R×R+.
If for anya˜,b˜∈R+×R and t,x,s∈0,T×R×R+, we have(24)liminfς⟶0Vπ∗t,x,s−Vπςt,x,sς≥0,which holds; then, π∗t,x,s is called an equilibrium strategy, and the equilibrium value function Vt,x,s is given by(25)Vt,x,s=Vπ∗t,x,s.
As evidenced by many references such as Bjӧrk and Murgoci [9] and Yang et al. [13], the equilibrium strategy defined by Definition 2 is time-consistent. Therefore, in the following, we call the equilibrium strategy and the equilibrium value function as the optimal time-consistent strategy and the optimal value function. Thus, the AAI’s aim is to find an optimal time-consistent investment strategy to solve robust optimization problem (19).
For convenience, we define two notations:(26)C1,2,20,T×R×R+=φt,x,sfor∀0,T×R×R+,φxt,x,s,φxxt,x,s,φst,x,s,φsst,x,sarecontinuous,and the usual infinitesimal generator Aπ,θ for the wealth process (17) is given by(27)Aπ,θφt,x,s=φtt,x,s+r˜x+π1tμ2−r1+π2tr˜2+π2tθ1tσ1+π1tθ2tσ2sβφxt,x,s+12π12tσ22s2β+π22tσ12φxxt,x,s+μ2s+θ2tσ2sβ+1φst,x,s+12σ22s2β+2φss+π1tσ22ts2β+1φsxt,x,s,where φt,x,s∈C1,2,20,T×R×R+, π∈Π, and θ∈Θ.
To ensure the strategy obtained from robust MV problem (19) is optimal, we present the following verification theorem. The proof of this theorem is similar to that in Theorem 7.1 of Bjӧrk and Murgoci [9], Theorem 4.1 of Yang et al. [13], and other papers, so we omit it here.Theorem 1.
For robust MV problem (19), if there exist two functions Wt,x,s and gt,x,s satisfying the following extended HJB equation(28)supπ∈ΠinfQ∈QAπ,θWt,x,s−Aπ,θγ2g2t,x,s+γgt,x,sAπ,θgt,x,s+θ1t22ϕ1t+θ2t22ϕ2t=0,WT,x,s=x,(29)Aπ∗,θ∗gt,x,s=0,gT,x,s=x,(30)π∗,θ∗=argsupπ∈ΠinfQ∈QAπ,θWt,x,s−Aπ,θγ2g2t,x,s+γgt,x,sAπ,θgt,x,s+θ1t22ϕ1t+θ2t22ϕ2t,then Vt,x,s=Wt,x,s, Et,x,sQXTπ∗=gt,x,s, π∗ is the optimal robust time-consistent investment strategy for AAI, and θ∗ is the optimal probability scenario for the market.


## 4. The Solution to the Robust MV Problem

This section is devoted to solve robust MV problem (19). By solving HJB equations (28) and (29), we can derive the solution to robust MV problem (19). To solve robust MV problem (19), similar to Maenhout [34, 35] and Chen and Yang [36], this article assumes that ϕ1t=α1 and ϕ2t=α2, where α1 and α1 are nonnegative.Now, we give an explicit expression for HJB equation (28).According to (27), we obtain(31)Aπ,θγ2g2t,x,s=γggt+r˜x+π1tμ2−r1+π2tr˜2+π2tθ1tσ1+π1tθ2tσ2sβγggx+12π12tσ22s2β+π22tσ12γgx2+γggxx+μ2sγggs+12σ22s2β+2γgs2+γggss+π1tσ22s2β+1γgsgx+γggsx,γgt,x,sAπ,θgt,x,s=γggt+r˜x+π1tμ2−r1+π2tr˜2+π2tθ1tσ1+π1tθ2tσ2sβγggx+12π12tσ22s2β+π22tσ12γggxx+μ2sγggs+12σ22s2β+2γggss+π1tσ22ts2β+1γggsx,where gt,x,s is abbreviated as g and gt, gx, gxx, gs, gss, and gsx are the partial derivatives of g with respect to the corresponding variables.Then, HJB equation (28) can be more explicitly expressed as(32)supπ∈ΠinfQ∈QWt+r˜x+π1tμ2−r1+π2tr˜2+π2tθ1tσ+π1tθ2tσ2sβWx+12π12tσ22s2β+π22tσ12Wxx−γgx2+μ2s+θ2tσ2sβ+1Ws+12σ22s2β+2Wss−γgs2+π1tσ22s2β+1Wsx−γgsgx+θ12t2α1+θ22t2α2=0,where Wt,x,s is abbreviated as W and Wt, Wx, Wxx, Ws, Wss, and Wsx are the partial derivatives of W with respect to the corresponding variables.Theorem 2.
For robust MV problem (19), the optimal robust time-consistent investment strategies are given by(33)π1∗t=μ2−r1+2γβσ22B¯t+2α2βσ22Btα2+γσ22s2βe−r˜T−t,(34)π2∗t=r˜2α1+γσ12e−r˜T−t.The optimal probability scenarios for the market are given by(35)θ1∗t=−α1r˜2α1+γσ1,(36)θ2∗t=2βα2σ2Btsβ−α2μ2−r1+2γβσ22B¯t+2α2βσ22Btα2+γσ2sβ.The optimal value function is given by(37)Wt,x,s=xer˜T−t+Bts−2β+Ct,where Ct is given by (60), and B¯t and Bt are determined by the following ordinary differential equations (ODEs):(38)B′t−2μ2βBt−2σ22γβ2B¯2−2α2β2σ22B2t+μ2−r1+2βσ22γB¯t+α2Bt22α2+γσ22=0,B¯′t−2B¯tμ2β+4α2β2σ22Bt+1α2+γσ22μ2−r1+2σ22α2βB¯t+Bt×μ2−r1+2βσ22γB¯t+α2Bt−α2μ2−r1+2βσ22γB¯t+α2Bt2α2+γ2σ22=0,with boundary conditions BT=0 and B¯T=0.Proof.
According to the structure of the wealth dynamics (17) and the boundary conditions WT,x,s=gT,x,s=x, we will seek the solution to HJB equations (28) and (29) with the following parametric form:(39)Wt,x,s=Atx+Bts−2β+Ct,with boundary conditions(40)AT=1,BT=0,CT=0,gt,x,s=A¯tx+B¯ts−2β+C¯t,with boundary conditions(41)A¯T=1,B¯T=0,C¯T=0.
Then, the partial derivatives ofWt,x,s and gt,x,s are given by(42)Wt=A′tx+B′ts−2β+C′t,Wx=At,Wxx=0,Ws=−2βBts−2β−1,Wss=2β2β+1Bts−2β−2,Wsx=0,gt=A¯′tx+B¯′ts−2β+C¯′t,gx=A¯t,gxx=0,gs=−2βB¯ts−2β−1,gss=2β2β+1B¯ts−2β−2,gsx=0.
Substituting equations (39)–(42) into equation (32), we obtain(43)A′t+r˜Atx+B′t−2μ2βBt−2σ22γβ2B¯2s−2β+C′t+σ22β2β+1Bt+supπ∈ΠinfQ∈Qθ22t2α2+θ2tπ1tσ2sβAt−2σ2βBts−β+θ12t2α1+π2θ1tσ1At−12π12tγσ22s2βA¯2t+π1tμ2−r1At+2γσ22βA¯tB¯t−12π22tγσ12A¯2t+π2tr˜2At=0.
According to the first-order optimality condition,θ1t and θ2t, solving the minimization problem in equation (43), we obtain(44)θ1∗t=−σ1α1π2tAt,(45)θ2∗t=−σ2α2π1tsβAt+2α2βσ2s−βBt.
Plugging (44) and (45) into equation (43) yields(46)A′t+r˜Atx+B′t−2μ2βBt−2σ22γβ2B¯2−2α2β2σ22B2ts−2β+C′t+σ22β2β+1Bt+supπ∈Π−12π12α2A2t+γA¯2tσ22s2β+π1tμ2−r1At+2γσ22βA¯tB¯t+2α2σ22βAtBt−12π22α1A2t+γA¯2tσ12+π2tr˜2At=0.
According to the first-order optimality condition,π1∗t and π2∗t, solving the maximization problem in equation (46), we obtain(47)π1∗t=μ2−r1At+2γβσ22A¯tB¯t+2α2σ22βAtBtα2A2t+γA¯2tσ22ts2β,(48)π2∗t=r˜2Atα1A2t+γA¯2tσ12t.
Inserting (47) and (48) into equation (46) and after simplifying give(49)A′t+r˜Atx+B′t−2μ2βBt−2σ22γβ2B¯2−2α2β2σ22B2t+μ2−r1At+2γβσ22A¯tB¯t+2α2σ22βAtBt22α2A2t+γA¯2tσ22s−2β+C′t+σ22β2β+1Bt+r˜22A2t2α1A2t+γA¯2tσ12=0.
By substituting (40), (42), (44), (45), (47), and (48) into equation (29) yields(50)A¯′t+r˜A¯tx+B¯′t−2μ2βB¯t−4α2β2σ22BtB¯t+1α2A2t+γA¯2tσ22μ2−r1A¯t+2σ22α2βAtB¯t+A¯tBt×μ2−r1At+2γβσ22A¯tB¯t+2α2σ22βAtBt−α2AtA¯tμ2−r1At+2γβσ22A¯tB¯t+2α2σ22βAtBt2α2A2t+γA¯2t2σ22s−2β+C¯′t+σ22β2β+1B¯t+r˜22AtA¯tα1A2t+γA¯2tσ12−α1r˜22A3tA¯tα1A2t+γA¯2t2σ12=0.
Thus, we must solve the following ODEs:(51)A′t+r˜At=0,AT=1,(52)B′t−2μ2βBt−2σ22γβ2B¯2−2α2β2σ22B2t+μ2−r1At+2γβσ22A¯tB¯t+2α2σ22βAtBt22α2A2t+γA¯2tσ22=0,BT=0,(53)C′t+σ22β2β+1Bt+r˜22A2t2α1A2t+γA¯2tσ12=0,CT=0,(54)A¯′t+r˜A¯t=0,A¯T=1,(55)B¯′t−2μ2βB¯t−4α2β2σ22BtB¯t+1α2A2t+γA¯2tσ22μ2−r1A¯t+2σ22α2βAtB¯t+A¯tBt×μ2−r1At+2γβσ22A¯tB¯t+2α2σ22βAtBt−α2AtA¯tμ2−r1At+2γβσ22A¯tB¯t+2α2σ22βAtBt2α2A2t+γA¯2t2σ22=0,B¯T=0,(56)C¯′t+σ22β2β+1B¯t+r˜22AtA¯tα1A2t+γA¯2tσ12−α1r˜22A3tA¯tα1A2t+γA¯2t2σ12=0,C¯T=0.
First, solving equations (51) and (54), respectively, we obtain(57)At=er˜T−t,(58)A¯t=er˜T−t.
By substituting (57) and (58) into (52) and (55), we can obtain Bt and B¯t which are the solutions of ODEs (38).
Inserting (57), (58), and B¯t into equation (56), we obtain the solution to equation (56) which is given by(59)C¯t=γr˜22T−tα1+γ2σ12+σ22β2β+1∫tTB¯sds.
By substituting (57), (58), and Bt into equation (53), and solving it, we obtain(60)Ct=r˜222α1+γσ12T−t+σ22β2β+1∫tTBsds.
Inserting (57), (58), Bt, and B¯t into (47), we can obtain (33). By substituting (57) and (58) into (48), we can obtain (34). Plugging (33) into (45) and (34) into (44), we yield (36) and (35), respectively.
Based on Theorem2 and the notation setting (8), we have the following theorem.Theorem 3.
For the wealth process (7), the optimal robust time-consistent investment strategies to robust MV problem (19) are given by(61)u1∗t=μ2−r1+2γβσ22B¯t+2α2βσ22Btxα2+γσ22s2βe−r˜T−t,(62)u2∗t=r˜2xα1+γσ12e−r˜T−t+1.Remark 2.
From the expressions of optimal robust time-consistent investment strategies (61) and (62), we note that u1∗t is related to the appreciation rate μ2 and price volatility σ2 of the risky asset and the current wealth x, whereas u2∗t is related to the parameters μ1 and σ1 of the inflation level and the current wealth x. This difference between u1∗t and u2∗t is mainly because the inflation price process and the risky asset price process are independent in our model assumption.


## 5. Numerical Experiments

This section is devoted to providing the influences of model parameters on the optimal time-consistent investment strategy and utility loss function by numerical experiments. In the following, unless otherwise stated, we choose the basic parameter setting as given in Table1.Table 1
Values of model parameters.
μ1σ1r1μ2σ2r2γxTtβsα1α20.050.30.030.080.250.022101090.1221


### 5.1. The Influences of the Parameters onu1∗t

We first present the effects of model parameters onu1∗t, where u1∗t is determined by (61).Lets∈2,10; Figure 1 shows the effect of s on u1∗t. From Figure 1, it is evident that u1∗t is a decreasing function of s, which describes the price of the risky asset. Form the meaning of s, we know that larger s implies that the investment risk becomes greater. Therefore, as s increases, the AAI will wish to invest less in the risky asset.Figure 1
Effect ofs on u1∗t.Figure2 presents two curves about the effect of β on u1∗t, where parameter β∈0.1,0.6. From Figure 2, we can see that when s=2, u1∗t is a decreasing function in β, while if s=0.5, u1∗t is an increasing function in β. When s=2 (s=0.5), the larger β is, the greater (less) sβ will be and hence the greater (less) the risk will be; thus, the AAI will wish to invest less (greater) in the risky asset.Figure 2
Effect ofβ on u1∗t.Letμ2∈0.05,0.1; we present the effect of μ2 on u1∗t in Figure 3. From Figure 3, it is clear that u1∗t is increasing with respect to μ2, which is the appreciation rate of the risky asset. Larger μ2 implies that the expected income of the risky asset becomes greater. As a result, the AAI will wish to invest more in the risky asset.Figure 3
Effect ofμ2 on u1∗t.Letσ2∈0.2,0.6; Figure 4 displays the effect of σ2 on u1∗t. From Figure 4, we can observe that u1∗t is a decreasing function of σ2. Recall that σ2 is the price volatility; therefore, with the increase of σ2, the investment risk will increase. Thus, the AAI will wish to invest less in the risky asset.Figure 4
Effect ofσ2 on u1∗t.Figure5 illustrates the effect of the risk-free interest rate r1 on u1∗t, where r1∈0.02,0.06. This figure shows that u1∗t is a decreasing function with respect to r1. Larger r1 implies that the expected income of the risk-free asset becomes greater. A natural result is that the AAI will invest more in the risk-free asset. That is to say, the AAI will invest less in the risky asset.Figure 5
Effect ofr1 on u1∗t.Figure6 discloses that u1∗t is a decreasing function of α2, where α2∈1,6. Recall that α2 stands for the AAI’s ambiguity aversion. Larger α2 implies that the model uncertainty becomes greater. Hence, the AAI will invest less in the risky asset.Figure 6
Effect ofα2 on u1∗t.Figure7 shows the effect of γ on u1∗t, where γ∈2,8. From Figure 7, we see that u1∗t is a decreasing function with respect to γ. This is obvious. Since γ is the risk-aversion parameter, larger γ implies that the less aggressive the AAI will be. Therefore, the AAI will invest less in the risky asset.Figure 7
Effect ofγ on u1∗t.Figure8 reveals that u1∗t is increasing with respect to t, where t∈2,9. This implies that, as the termination time of the investment approaches, the AAI will invest more in the risky assets.Figure 8
Effect oft on u1∗t.


### 5.2. The Influences of the Parameters onu2∗t

Now, we illustrate the effects of model parameters on the optimal robust time-consistent investment strategyu2∗t, where u2∗t is determined by (62).Differentiating (62) with respect to μ1, we obtain(63)∂u2∗t∂μ1=1xα1+γσ12e−r˜T−t>0,that is, u2∗t is increasing with respect to μ1. This implies that, as the interest rate of the inflation-indexed bond increases, the AAI should retain more investments in the inflation-indexed bond.Differentiating (62) with respect to σ1, we have(64)∂u2∗t∂σ1=−2r2+μ1−r1xα1+γσ13e−r˜T−t<0,so we know that u2∗t is decreasing with respect to σ1. Recall that σ1 is the volatility of the inflation-indexed bond. Therefore, larger σ1 implies that the investment risk in the inflation-indexed bond becomes larger. Hence, the AAI retains less investments in the inflation-indexed bond.Differentiating (62) with respect to r1, we obtain(65)∂u2∗t∂r1=−1xα1+γσ12e−r˜T−t<0,which implies that u2∗t is decreasing with respect to r1. The reason is similar to what we have explained in Figure 5.In the following, we analyze the effect ofr2 on u2∗t by the numerical experiment. We set μ1=0.02,σ1=0.04,α1=10,γ=2,x=10,T=10,t=5,r2∈0.01,0.03, and the result is shown in Figure 9. From Figure 9, we can observe that u2∗t is an increasing function with respect to r2. Recall that r2 stands for the real interest rate of the inflation-indexed bond. Hence, the larger r2 is, the more the AAI will invest in the inflation-indexed bond.Figure 9
Effect ofr2 on u2∗t.


### 5.3. The Influences of Model Parameters on Utility Loss Function

In Sections5.1 and 5.2, we studied the case under the robust model. This part uses numerical experiments to investigate the influences of model parameters on utility loss function.First, we give some results under the ambiguity-neutral case.For an ambiguity-neutral optimization problem, all admissible investment strategiesπ¯t are denoted by Π¯. Then, the wealth process becomes(66)dXtπ¯=r˜Xtπ¯+π¯1tμ2−r1+π¯2tr˜2dt+π¯1tσ2SβtdW2t+π¯2tσ1dW1t,and the objective function is as follows:(67)supπ¯∈Π¯Et,x,sXTπ¯−γ2Vart,x,sXTπ¯.Similar to what we have proved in Theorem2, we can obtain the following theorem.Theorem 4.
For problem (67), the optimal time-consistent investment strategies are given by(68)π¯1∗t=μ2−r1+2γβσ22B2tγσ22s2βe−r˜T−t,(69)π¯2∗t=r˜2γσ12e−r˜T−t.
Furthermore, the optimal value function is given by(70)W1t,x,s=xer˜T−t+B1ts−2β+C1t,where(71)B1t=e−2μ2βT−t∫tTf1se2μ2βT−sds,B2t=μ2−r122r1βγσ221−e−2r1βT−t,f1t=2βμ2−r1B2t+μ2−r122γσ22,C1t=r˜222γσ12T−t+σ22β2β+1∫tTB1sds.
Based on Theorems2 and 4, we can define the utility loss function as(72)Lt=1−W1t,x,sWt,x,s.
In the following, we investigate the effects of model parameters onLt.(i)
The influence of ambiguity aversion on the utility loss:Letα2=4 and α1=2,3,4,5,6,7,8; we can obtain W1t,x,s=10.2370. The values of Wt,x,s and Lt are presented in Table 2. From Table 2, we know that the utility loss function is a decreasing function in α1, where α1 stands for the ambiguity aversion which arises from the inflation-indexed bond. By comparing the AAI’s utility loss at different α1, we find that the more ambiguity averse the AAI is, the less conservative investment strategy the AAI will be (because Wt,x,s is decreasing in α1). Because AAI's investment is robust, he will invest more money in the risk-free asset, so AAI will suffer less utility loss.Letα1=4 and α2=2,3,4,5,6,7,8; we have W1t,x,s=10.2370. The values of Wt,x,s and Lt are presented in Table 3. From Table 3, it is evident that the utility loss function is an increasing function in α2, where α2 stands for the ambiguity aversion which arises from the risky asset.(ii)
The influence of market parameters on the utility loss:Figure10 discloses that Lt is a decreasing function with respect to μ2, where μ2∈0.05,0.1. This indicates that the utility loss increases as the appreciation rate of the risky asset diminishes.Figure11 reveals that Lt is an increasing function with regard to σ2, where σ2∈0.2,0.6. Larger σ2 implies that the investment risk becomes greater, and thus, the AAI will suffer from more utility losses.Lets∈0.5,1.5; Figure 12 gives the influence of parameter s on the utility loss. From Figure 12, we can observe that Lt is an increasing function of s. The larger the s is, the larger the price volatility will be, that is, the greater the risk will be, and hence, the AAI will suffer from more utility losses.Letσ1∈0.2,0.6; Figure 13 shows that the utility loss is a decreasing function of σ1. Comparing Figures 11 with 13, it is clear that the effect of the volatilities of the risky asset and the inflation on the utility loss is opposite.Table 2
Effect ofα1 on Wt,x,s and Lt.
α12345678Wt,x,s11.721711.721211.720911.720711.720511.720411.7202Lt0.126660.126630.12660.126580.126570.126560.12655Table 3
Effect ofα2 on Wt,x,s and Lt.
α22345678Wt,x,s11.030711.038611.042911.045511.047211.048411.0492Lt0.07190.07260.07300.07320.07330.07340.0735Figure 10
Effect ofμ2 on Lt.Figure 11
Effect ofσ2 on Lt.Figure 12
Effect ofs on Lt.Figure 13
Effect ofσ1 on Lt.


## 5.1. The Influences of the Parameters onu1∗t

We first present the effects of model parameters onu1∗t, where u1∗t is determined by (61).Lets∈2,10; Figure 1 shows the effect of s on u1∗t. From Figure 1, it is evident that u1∗t is a decreasing function of s, which describes the price of the risky asset. Form the meaning of s, we know that larger s implies that the investment risk becomes greater. Therefore, as s increases, the AAI will wish to invest less in the risky asset.Figure 1
Effect ofs on u1∗t.Figure2 presents two curves about the effect of β on u1∗t, where parameter β∈0.1,0.6. From Figure 2, we can see that when s=2, u1∗t is a decreasing function in β, while if s=0.5, u1∗t is an increasing function in β. When s=2 (s=0.5), the larger β is, the greater (less) sβ will be and hence the greater (less) the risk will be; thus, the AAI will wish to invest less (greater) in the risky asset.Figure 2
Effect ofβ on u1∗t.Letμ2∈0.05,0.1; we present the effect of μ2 on u1∗t in Figure 3. From Figure 3, it is clear that u1∗t is increasing with respect to μ2, which is the appreciation rate of the risky asset. Larger μ2 implies that the expected income of the risky asset becomes greater. As a result, the AAI will wish to invest more in the risky asset.Figure 3
Effect ofμ2 on u1∗t.Letσ2∈0.2,0.6; Figure 4 displays the effect of σ2 on u1∗t. From Figure 4, we can observe that u1∗t is a decreasing function of σ2. Recall that σ2 is the price volatility; therefore, with the increase of σ2, the investment risk will increase. Thus, the AAI will wish to invest less in the risky asset.Figure 4
Effect ofσ2 on u1∗t.Figure5 illustrates the effect of the risk-free interest rate r1 on u1∗t, where r1∈0.02,0.06. This figure shows that u1∗t is a decreasing function with respect to r1. Larger r1 implies that the expected income of the risk-free asset becomes greater. A natural result is that the AAI will invest more in the risk-free asset. That is to say, the AAI will invest less in the risky asset.Figure 5
Effect ofr1 on u1∗t.Figure6 discloses that u1∗t is a decreasing function of α2, where α2∈1,6. Recall that α2 stands for the AAI’s ambiguity aversion. Larger α2 implies that the model uncertainty becomes greater. Hence, the AAI will invest less in the risky asset.Figure 6
Effect ofα2 on u1∗t.Figure7 shows the effect of γ on u1∗t, where γ∈2,8. From Figure 7, we see that u1∗t is a decreasing function with respect to γ. This is obvious. Since γ is the risk-aversion parameter, larger γ implies that the less aggressive the AAI will be. Therefore, the AAI will invest less in the risky asset.Figure 7
Effect ofγ on u1∗t.Figure8 reveals that u1∗t is increasing with respect to t, where t∈2,9. This implies that, as the termination time of the investment approaches, the AAI will invest more in the risky assets.Figure 8
Effect oft on u1∗t.


## 5.2. The Influences of the Parameters onu2∗t

Now, we illustrate the effects of model parameters on the optimal robust time-consistent investment strategyu2∗t, where u2∗t is determined by (62).Differentiating (62) with respect to μ1, we obtain(63)∂u2∗t∂μ1=1xα1+γσ12e−r˜T−t>0,that is, u2∗t is increasing with respect to μ1. This implies that, as the interest rate of the inflation-indexed bond increases, the AAI should retain more investments in the inflation-indexed bond.Differentiating (62) with respect to σ1, we have(64)∂u2∗t∂σ1=−2r2+μ1−r1xα1+γσ13e−r˜T−t<0,so we know that u2∗t is decreasing with respect to σ1. Recall that σ1 is the volatility of the inflation-indexed bond. Therefore, larger σ1 implies that the investment risk in the inflation-indexed bond becomes larger. Hence, the AAI retains less investments in the inflation-indexed bond.Differentiating (62) with respect to r1, we obtain(65)∂u2∗t∂r1=−1xα1+γσ12e−r˜T−t<0,which implies that u2∗t is decreasing with respect to r1. The reason is similar to what we have explained in Figure 5.In the following, we analyze the effect ofr2 on u2∗t by the numerical experiment. We set μ1=0.02,σ1=0.04,α1=10,γ=2,x=10,T=10,t=5,r2∈0.01,0.03, and the result is shown in Figure 9. From Figure 9, we can observe that u2∗t is an increasing function with respect to r2. Recall that r2 stands for the real interest rate of the inflation-indexed bond. Hence, the larger r2 is, the more the AAI will invest in the inflation-indexed bond.Figure 9
Effect ofr2 on u2∗t.


## 5.3. The Influences of Model Parameters on Utility Loss Function

In Sections5.1 and 5.2, we studied the case under the robust model. This part uses numerical experiments to investigate the influences of model parameters on utility loss function.First, we give some results under the ambiguity-neutral case.For an ambiguity-neutral optimization problem, all admissible investment strategiesπ¯t are denoted by Π¯. Then, the wealth process becomes(66)dXtπ¯=r˜Xtπ¯+π¯1tμ2−r1+π¯2tr˜2dt+π¯1tσ2SβtdW2t+π¯2tσ1dW1t,and the objective function is as follows:(67)supπ¯∈Π¯Et,x,sXTπ¯−γ2Vart,x,sXTπ¯.Similar to what we have proved in Theorem2, we can obtain the following theorem.Theorem 4.
For problem (67), the optimal time-consistent investment strategies are given by(68)π¯1∗t=μ2−r1+2γβσ22B2tγσ22s2βe−r˜T−t,(69)π¯2∗t=r˜2γσ12e−r˜T−t.
Furthermore, the optimal value function is given by(70)W1t,x,s=xer˜T−t+B1ts−2β+C1t,where(71)B1t=e−2μ2βT−t∫tTf1se2μ2βT−sds,B2t=μ2−r122r1βγσ221−e−2r1βT−t,f1t=2βμ2−r1B2t+μ2−r122γσ22,C1t=r˜222γσ12T−t+σ22β2β+1∫tTB1sds.
Based on Theorems2 and 4, we can define the utility loss function as(72)Lt=1−W1t,x,sWt,x,s.
In the following, we investigate the effects of model parameters onLt.(i)
The influence of ambiguity aversion on the utility loss:Letα2=4 and α1=2,3,4,5,6,7,8; we can obtain W1t,x,s=10.2370. The values of Wt,x,s and Lt are presented in Table 2. From Table 2, we know that the utility loss function is a decreasing function in α1, where α1 stands for the ambiguity aversion which arises from the inflation-indexed bond. By comparing the AAI’s utility loss at different α1, we find that the more ambiguity averse the AAI is, the less conservative investment strategy the AAI will be (because Wt,x,s is decreasing in α1). Because AAI's investment is robust, he will invest more money in the risk-free asset, so AAI will suffer less utility loss.Letα1=4 and α2=2,3,4,5,6,7,8; we have W1t,x,s=10.2370. The values of Wt,x,s and Lt are presented in Table 3. From Table 3, it is evident that the utility loss function is an increasing function in α2, where α2 stands for the ambiguity aversion which arises from the risky asset.(ii)
The influence of market parameters on the utility loss:Figure10 discloses that Lt is a decreasing function with respect to μ2, where μ2∈0.05,0.1. This indicates that the utility loss increases as the appreciation rate of the risky asset diminishes.Figure11 reveals that Lt is an increasing function with regard to σ2, where σ2∈0.2,0.6. Larger σ2 implies that the investment risk becomes greater, and thus, the AAI will suffer from more utility losses.Lets∈0.5,1.5; Figure 12 gives the influence of parameter s on the utility loss. From Figure 12, we can observe that Lt is an increasing function of s. The larger the s is, the larger the price volatility will be, that is, the greater the risk will be, and hence, the AAI will suffer from more utility losses.Letσ1∈0.2,0.6; Figure 13 shows that the utility loss is a decreasing function of σ1. Comparing Figures 11 with 13, it is clear that the effect of the volatilities of the risky asset and the inflation on the utility loss is opposite.Table 2
Effect ofα1 on Wt,x,s and Lt.
α12345678Wt,x,s11.721711.721211.720911.720711.720511.720411.7202Lt0.126660.126630.12660.126580.126570.126560.12655Table 3
Effect ofα2 on Wt,x,s and Lt.
α22345678Wt,x,s11.030711.038611.042911.045511.047211.048411.0492Lt0.07190.07260.07300.07320.07330.07340.0735Figure 10
Effect ofμ2 on Lt.Figure 11
Effect ofσ2 on Lt.Figure 12
Effect ofs on Lt.Figure 13
Effect ofσ1 on Lt.


## 6. Conclusions

We have investigated a robust time-consistent MV strategy selection under the inflation risk for an AAI. The AAI doubts the model setting under the original probability measure. Then, he seeks a set of alternative probability measures, which are absolutely continuous to the original probability measure. Furthermore, we modify the model setting under the alternative probability measures. We form the AAI’s objective under the framework of the robust MV criterion. As we all know, this problem is time-inconsistent. To solve this time-inconsistent problem, we seek the equilibrium strategy from the game framework. The equilibrium strategy is time-consistent. By using the stochastic optimal control technique, we derive the closed-form solutions for the optimal time-consistent investment strategy and the optimal value function. Finally, the influences of model parameters on the optimal time-consistent investment strategy and utility loss function are examined through numerical experiments.Compared with Zeng et al. [12], Zheng et al. [28], Maenhout [34, 35], and other studies about the robust MV problem, the inflation risk and the CEV model are simultaneously introduced in this article. In addition, the influence of the model parameters of the inflation on the optimal time-consistent investment strategy and the utility loss are systematic analysis by numerical experiments.It may be interesting to extend the current framework to that with dividend and state-dependent risk-aversion function or that in a partial information market in the future research. For these future research topics, we may need to adopt the methods used in Bjӧk et al. [14], Yu et al. [37], and Raychaudhury and Pal [38].



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