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

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.


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 continuoustime 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. e aforementioned work under the MV criterion is time inconsistency. at is to say, the optimal strategy made at time t 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. us, 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. e financial market includes one risk-free asset, one risky asset, and one inflation-indexed bond. e 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. en, the new model setting is obtained under the new probability measure. e 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. e 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 timeconsistent 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. e 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) e influences of model parameters on the optimal time-consistent investment strategy and utility loss function are systematically examined through numerical experiments.
e remainder of this article is arranged as follows. In Section 2, 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 timeconsistent investment strategy and utility loss function are examined through numerical experiments. e final section summarizes this article.

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 space (Ω, F, F t , P). We assume that F ≔ F t , t ≥ 0 is right-continuous and is complete with respect to P. F t stands for the information acquired by the investor up to time t. We assume that there are no market frictions in trading.

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 level P 0 (t) satisfies the following process: where μ 1 > 0 represents the instantaneous expected rate of the inflation, σ 1 > 0 stands for the volatility of the inflation, and W 1 (t) 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. e price process P 1 (t) of the risk-free asset is given by Here, r 1 > 0 represents the interest rate of the risk-free asset. e price process S(t) of the risky asset satisfies the following CEV model: Here, μ 2 ≥ r 1 stands for the appreciation rate, σ 2 > 0 represents the price volatility, W 2 (t) represents a standard Brownian motion, and β is the constant elasticity parameter. We assume W 1 (t) and W 2 (t) are mutually independent.

Remark 1.
e parameter β 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.
e 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. en, 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].
e third asset to invest in is the inflation-indexed bond. Similar to Kwak and Lim [19], the price P(t) of the inflationindexed bond is given by Here, r 2 > 0 represents the real interest rate, and r 2 + μ 1 > r 1 stands for the appreciation rate.

Inflation-Adjusted Wealth Process.
Let u 1 (t) and u 2 (t), 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 u 1 (t) and u 2 (t) as control strategies; we denote them as u(t) � (u 1 (t), u 2 (t)). For each strategy u(t), the wealth process R u t can be described as Denote By applying Itô's formula, X u t can be described as Here, X u t is the real wealth with stripping out inflation. To facilitate solving the optimization problem in Section 3, we define the following notations: en, the real wealth X π t can be described as dX π t � rX π t + π 1 (t) μ 2 − r 1 + π 2 (t)r 2 dt + π 1 (t)σ 2 S β (t)dW 2 (t) + π 2 (t)σ 1 dW 1 (t).

Problem Formulation
MV investment strategy selection problem as a classic optimization problem can be given as follows: Here x 0 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: where c > 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: e target is to develop the corresponding equilibrium investment strategy, which is timeconsistent.
Problem (12) is the traditional MV problem, where the investor is ambiguity-neutral. at 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,

Mathematical Problems in Engineering
Now, we define an admissible strategy.

Definition 1.
For any fixed t ∈ [0, T], an investment strategy π(t) � (π 1 (t), π 2 (t)) is called an admissible strategy for the ambiguity-averse investor (AAI) if it satisfies the following: (9) with respect to π(t) has a unique strong solution We denote all such admissible investment strategies on the time interval [0, T] as Π.
To transform model (9) from the probability measure P to the probability measure Q, we define a process θ(t) � (θ 1 (t), θ 2 (t)) | t ∈ [0, T] , which satisfies the following: en, Z θ (t) is a martingale with respect to P. In the following, we define a new probability measure Q, which is given by From the definition of Q, it is clear that Q is absolutely continuous to P.
For any θ ∈ Θ, we, respectively, define two new processes According to Girsanov's theorem, W Q 1 (t) and W Q 2 (t) are standard Brownian motions with respect to Q. Furthermore, the wealth process (9) under Q can be rewritten as and corresponding CEV model (3) becomes In the following, we modify MV problem (12) under the worst-case scenario. rough modifying MV problem (12), we shall deal with a robust time-consistent MV strategy selection problem as follows: Here, where ϕ 1 (t) and ϕ 2 (t) are nonnegative and stand for the investor's ambiguity aversion with respect to the model under P. e larger ϕ 1 (t) and ϕ 2 (t) are, the more the model under Q will deviate from the model under P. erefore, AAI's ambiguity aversion is an increasing function of ϕ 1 (t) and ϕ 2 (t).
To embody some good properties of the model under P, AAI deviation from P is penalized by the first two terms in (20). e penalty terms depend on the relative entropy. e increase in relative entropy from t to t + dt equals e 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.
we choose an admissible strategy π * (t, x, s) ∈ Π. en, through choosing three real numbers a ∈ R + , b ∈ R, and ς > 0, we define a new strategy by which holds; then, π * (t, x, s) is called an equilibrium strategy, and the equilibrium value function V(t, x, s) is given by 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. erefore, in the following, we call the equilibrium strategy and the equilibrium value function as the optimal time-consistent strategy and the optimal value function. us, the AAI's aim is to find an optimal time-consistent investment strategy to solve robust optimization problem (19).
To ensure the strategy obtained from robust MV problem (19) is optimal, we present the following verification theorem. e proof of this theorem is similar to that in eorem 7.1 of Bjӧrk and Murgoci [9], eorem 4.1 of Yang et al. [13], and other papers, so we omit it here. (19), if there exist two functions W(t, x, s) and g(t, x, s) satisfying the following extended HJB equation

Theorem 1. For robust MV problem
x, s), π * is the optimal robust time-consistent investment strategy for AAI, and θ * is the optimal probability scenario for the market.

Mathematical Problems in Engineering
where g(t, x, s) is abbreviated as g and g t , g x , g xx , g s , g ss , and g sx are the partial derivatives of g with respect to the corresponding variables.
en, HJB equation (28) can be more explicitly expressed as where W(t, x, s) is abbreviated as W and W t , W x , W xx , W s , W ss , and W sx 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 e optimal probability scenarios for the market are given by e optimal value function is given by where C(t) is given by (60), and B(t) and B(t) are determined by the following ordinary differential equations (ODEs): with boundary conditions B(T) � 0 and B(T) � 0.
Proof. According to the structure of the wealth dynamics (17) and the boundary conditions W(T, x, s) � g(T, x, s) � x, we will seek the solution to HJB equations (28) and (29) with the following parametric form: with boundary conditions with boundary conditions en, the partial derivatives of W(t, x, s) and g(t, x, s) are given by Substituting equations (39)-(42) into equation (32), we obtain 6 Mathematical Problems in Engineering According to the first-order optimality condition, θ 1 (t) and θ 2 (t), solving the minimization problem in equation (43), we obtain Plugging (44) and (45) into equation (43) yields According to the first-order optimality condition, π * 1 (t) and π * 2 (t), solving the maximization problem in equation (46), we obtain
Based on eorem 2 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 Remark 2. From the expressions of optimal robust timeconsistent investment strategies (61) and (62), we note that u * 1 (t) is related to the appreciation rate μ 2 and price volatility σ 2 of the risky asset and the current wealth x, whereas u * 2 (t) is related to the parameters μ 1 and σ 1 of the inflation level and the current wealth x. is difference between u * 1 (t) and u * 2 (t) is mainly because the inflation price process and the risky asset price process are independent in our model assumption.

Numerical Experiments
is 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 Table 1.

5.1.
e Influences of the Parameters on u * 1 (t). We first present the effects of model parameters on u * 1 (t), where u * 1 (t) is determined by (61). Let s ∈ [2, 10]; Figure 1 shows the effect of s on u * 1 (t). From Figure 1, it is evident that u * 1 (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. erefore, as s increases, the AAI will wish to invest less in the risky asset. Figure 2 presents two curves about the effect of β on u * 1 (t), where parameter β ∈ [0.1, 0.6]. From Figure 2, we can see that when s � 2, u * 1 (t) is a decreasing function in β, while if s � 0.5, u * 1 (t) is an increasing function in β. When s � 2 (s � 0.5), the larger β is, the greater (less) s β will be and hence the greater (less) the risk will be; thus, the AAI will wish to invest less (greater) in the risky asset.
Let μ 2 ∈ [0.05, 0.1]; we present the effect of μ 2 on u * 1 (t) in Figure 3. From Figure 3, it is clear that u * 1 (t) is increasing with respect to μ 2 , which is the appreciation rate of the risky asset. Larger μ 2 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.
Let σ 2 ∈ [0.2, 0.6]; Figure 4 displays the effect of σ 2 on u * 1 (t). From Figure 4, we can observe that u * 1 (t) is a decreasing function of σ 2 . Recall that σ 2 is the price volatility; therefore, with the increase of σ 2 , the investment risk will increase. us, the AAI will wish to invest less in the risky asset. Figure 5 illustrates the effect of the risk-free interest rate r 1 on u * 1 (t), where r 1 ∈ [0.02, 0.06]. is figure shows that u * 1 (t) is a decreasing function with respect to r 1 . Larger r 1 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. at is to say, the AAI will invest less in the risky asset. Figure 6 discloses that u * 1 (t) is a decreasing function of α 2 , where α 2 ∈ [1, 6]. Recall that α 2 stands for the AAI's ambiguity aversion. Larger α 2 implies that the model uncertainty becomes greater. Hence, the AAI will invest less in the risky asset. Figure 7 shows the effect of c on u * 1 (t), where c ∈ [2,8]. From Figure 7, we see that u * 1 (t) is a decreasing function with respect to c. is is obvious. Since c is the risk-aversion parameter, larger c implies that the less aggressive the AAI will be. erefore, the AAI will invest less in the risky asset. Figure 8 reveals that u * 1 (t) is increasing with respect to t, where t ∈ [2,9]. is implies that, as the termination time of the investment approaches, the AAI will invest more in the risky assets.

e Influences of the Parameters on u *
2 (t). Now, we illustrate the effects of model parameters on the optimal robust time-consistent investment strategy u * 2 (t), where u * 2 (t) is determined by (62). Differentiating (62) with respect to μ 1 , we obtain zu * 2 (t) that is, u * 2 (t) is increasing with respect to μ 1 . is 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 σ 1 , we have so we know that u * 2 (t) is decreasing with respect to σ 1 . Recall that σ 1 is the volatility of the inflation-indexed bond. erefore, larger σ 1 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 r 1 , we obtain which implies that u * 2 (t) is decreasing with respect to r 1 . e reason is similar to what we have explained in Figure 5.
In the following, we analyze the effect of r 2 on u * 2 (t) by the numerical experiment. We set μ 1 � 0.02, σ 1 � 0.04, α 1 � 10, c � 2, x � 10, T � 10, t � 5, r 2 ∈ [0.01, 0.03], and the result is shown in Figure 9. From Figure 9, we can observe that u * 2 (t) is an increasing function with respect to r 2 . Recall that r 2 stands for the real interest rate of the inflationindexed bond. Hence, the larger r 2 is, the more the AAI will invest in the inflation-indexed bond.

5.3.
e Influences of Model Parameters on Utility Loss Function. In Sections 5.1 and 5.2, we studied the case under the robust model. is 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 strategies π(t) are denoted by Π.
en, the wealth process becomes and the objective function is as follows: Similar to what we have proved in eorem 2, we can obtain the following theorem.

Theorem 4.
For problem (67), the optimal time-consistent investment strategies are given by Furthermore, the optimal value function is given by where Based on eorems 2 and 4, we can define the utility loss function as In the following, we investigate the effects of model parameters on L(t).  investment is robust, he will invest more money in the risk-free asset, so AAI will suffer less utility loss.  Table 3. From Table 3, it is evident that the utility loss function is an increasing function in α 2 , where α 2 stands for the ambiguity aversion which arises from the risky asset. (ii) e influence of market parameters on the utility loss: Figure 10 discloses that L(t) is a decreasing function with respect to μ 2 , where μ 2 ∈ [0.05, 0.1]. is indicates that the utility loss increases as the appreciation rate of the risky asset diminishes. Figure 11 reveals that L(t) is an increasing function with regard to σ 2 , where σ 2 ∈ [0.2, 0.6]. Larger σ 2 implies that the investment risk becomes greater, and thus, the AAI will suffer from more utility losses. Let s ∈ [0.5, 1.5]; Figure 12 gives the influence of parameter s on the utility loss. From Figure 12, we can observe that L(t) is an increasing function of s. e 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. Let σ 1 ∈ [0.2, 0.6]; Figure 13 shows that the utility loss is a decreasing function of σ 1 . 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.

Conclusions
We have investigated a robust time-consistent MV strategy selection under the inflation risk for an AAI. e AAI doubts the model setting under the original probability measure. en, 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. e 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].

Data Availability
e data used to support the findings of this study are available upon request to the author.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.