DYNAMIC MEAN-LPM and MEAN-CVaR PORTFOLIO OPTIMIZATION IN CONTINUOUS-TIME This research work was partially supported by Natural Science Foundation of China under grant 71201102, by Ph.D. Programs Foundation of Ministry of Education of China under grand 20120073120037, and by Hong Kong Research Grants Council under grants CUHK 414513 and CUHK414610. The third author is grateful to the support from Patrick Huen Wing Ming Professorship of Systems Engineering & Engineering Management.

DYNAMIC MEAN-LPM and MEAN-CVaR PORTFOLIO OPTIMIZATION IN CONTINUOUS-TIME thanks: This research work was partially supported by Natural Science Foundation of China under grant 71201102, by Ph.D. Programs Foundation of Ministry of Education of China under grand 20120073120037, and by Hong Kong Research Grants Council under grants CUHK 414513 and CUHK414610. The third author is grateful to the support from Patrick Huen Wing Ming Professorship of Systems Engineering & Engineering Management.

Jianjun Gao Department of Automation, Shanghai Jiao Tong University, Shanghai, China.    Ke Zhou Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong.    Duan LI Corresponding author. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong. E-mail: dli@se.cuhk.edu.hk.    Xiren Cao Department of Automation, Shanghai Jiao Tong University, Shanghai, China.
Abstract

Instead of controlling “symmetric” risks measured by central moments of investment return or terminal wealth, more and more portfolio models have shifted their focus to manage “asymmetric” downside risks that the investment return is below certain threshold. Among the existing downside risk measures, the lower-partial moments (LPM) and conditional value-at-risk (CVaR) are probably most promising. In this paper we investigate the dynamic mean-LPM and mean-CVaR portfolio optimization problems in continuous-time, while the current literature has only witnessed their static versions. Our contributions are two-fold, in both building up tractable formulations and deriving corresponding analytical solutions. By imposing a limit funding level on the terminal wealth, we conquer the ill-posedness exhibited in the class of mean-downside risk portfolio models. The limit funding level not only enables us to solve both dynamic mean-LPM and mean-CVaR portfolio optimization problems, but also offers a flexibility to tame the aggressiveness of the portfolio policies generated from such mean - downside risk models. More specifically, for a general market setting, we prove the existence and uniqueness of the Lagrangian multiplies, which is a key step in applying the martingale approach, and establish a theoretical foundation for developing efficient numerical solution approaches. Moreover, for situations where the opportunity set of the market setting is deterministic, we derive analytical portfolio policies for both dynamic mean-LPM and mean-CVaR formulations.

Key words. Dynamic mean - downside risk portfolio optimization, lower-partial moments (LPM), conditional value-at-risk portfolio (CVaR), stochastic control, martingale approach.

AMS subject classifications. 91G10, 91G80, 91G60

1 Introduction

The mean-variance (MV) formulation pioneered by Markowitz [26] sixty years ago has laid the foundation of modern portfolio theory. Most importantly, the mean-variance model captures the essential multiobjective nature between the two conflicting goals in portfolio selection, i.e., between maximizing the investment return and minimizing the investment risk. As a natural generalization of the mean-variance analysis, the framework of mean-risk trade-off analysis has become a standard in portfolio management. Under the framework of mean-risk trade-off analysis, a risk measure always serves a purpose to map investment uncertainty to a quantitative level such that trade-off can be computed explicitly against the expected investment return. Such a straightforward appealing approach of risk management is in general more favored by both practitioners in financial industry and researchers in academic field, when compared to the more abstract, albeit more mathematically rigorous, expected utility maximization framework. However, selecting an appropriate risk measure is essentially not only a science, but also an art.

While the variance term penalizes uncertainties on both sides of the mean, numerous downside risk measures have been proposed in the last half century to quantify the risk that the investment return is below certain target. Among these downside risk measures, the lower-partial moments (LPM) proposed by Fishbburn [14] form one most important class with prominent features. The LPM enables us to represent a general form of downside risk measures with two parameters, the benchmark level , which is set by the investor himself, and the order of the moments, , which represents the risk attitude of the investor. Due to the freedom offered by different combinations of the pair and , we can adopt LPM to pursue different investment goals in portfolio optimization. For example, setting in LMP yields the shortfall probability, which is also equivalent to the safety-first rule proposed by Roy [32]; Setting gives rise to the risk measure of the expected regret (ER) (see Dembo [11]); and setting leads to the risk measure of semideviation below the target, or the semivariance if is set as the expected terminal wealth. Bawa and Lindenberg [6] show that LPMs associated with , 1, or 2 correspond to the first, second or third degree stochastic dominance, respectively. Compared with the variance, the LPM is more consistent with the classical utility theory and the rule of stochastic dominance (see e.g., [27]). Konno et al. [22] demonstrate the prominence of LPM in the practice of portfolio management via empirical tests. Zhu et al. further consider robust portfolio selection under LPM risk measures [38].

The Value-at-Risk (VaR), defined as the threshold point with a specified exceeding probability of great loss, becomes popular in the financial industry since the mid 90s. However, the VaR has been widely criticized for some of its undesired properties. More specifically, VaR fails to satisfy the axiomatic system of coherent risk measures proposed by Artzner et al. [3]. Most critically, the non-convexity of VaR leads to some difficulty in solving the corresponding portfolio optimization problem. On the other hand, the conditional Value-at-Risk (CVaR), also known as the expected shortfall, is defined as the expected value of the loss exceeding the VaR [30]. CVaR possesses several good properties, such as convexity, monotonicity and homogeneity. Rockafellar and Uryasev [30] [31] prove that CVaR can be computed by solving an auxiliary linear programming problem in which the VaR needs not to be known in advance. After the fundamental work of Rockafellar and Uryasev ([30] [31]), CVaR has been widely applied in various applications of portfolio selection and risk management, e.g., derivative portfolio [1], credit risk optimization [2], and robust portfolio management [36].

Almost all the mean-downside risk portfolio optimization models studied in the above literature have been confined to static settings, from which the derived portfolio policy is of a buy-and-hold nature. Without a doubt, such a class of static models is not suitable for investment problems with a long investment horizon. The past decade has witnessed some research works that investigate mean-CVaR portfolio optimization using stochastic programming approach [12] [13] [17]. As stochastic programming formulations adopt both discrete time and discrete state in their model settings, this kind of models with discrete states suffers from a heavy computational burden, and can only deal with two - or three - stage problems. Within dynamic mean-risk portfolio optimization models, the most matured development seems to lie in the subject of dynamic mean-variance (MV) portfolio optimization. Although the mean-variance analysis starts the area of portfolio selection, its extension to a dynamic MV version has been blocked for almost four decades, due to the nonseparability of the variance term in the sense of dynamic programming. After Li and Ng [23] and Zhou and Li [35] derive the explicit portfolio policies, respectively, for discrete-time and continuous-time MV portfolio selection formulations, by using the embedding scheme, the dynamic MV models has been developed by leaps and bounds, see, for examples, [24] [25] [37] [19]. Recently, the subject of time consistency in dynamic MV portfolio optimization has been attracting increasing attention (see, e.g., [4] [9] [8]). Although the mean-downside risk models seem to be a natural extension of dynamic MV models, Jin et al. [18] show that a general class of mean-downside risk portfolio optimization models under a continuous-time setting is ill-posed in the sense that the optimal value cannot be achieved. Besides such a negative result, there do exist some research works related to the continuous-time portfolio selection problems in which the downside-risk measure plays a role. For example, Basak and Shapiro [5] consider the continuous-time utility maximization model with a VaR constraint. By using the stochastic control approach, Yiu [34] study a problem similar to [5]. However, the VaR risk constraint in Yiu [34] is defined over the entire investment process. Gundel and Weber [16] extend the VaR risk constraint to a shortfall risk constraint. Recently, Chiu et al. [10] solve the dynamic asset-liability management problem under the safety-first criteria, which can be regarded as the shortfall probability measure.

We consider in this paper the mean-downside risk portfolio optimization problem in a continuous-time setting. More specifically, we investigate both the dynamic mean-LPM and mean-CVaR portfolio optimization problems. In recognizing the ill-posedness of such problems (see, e.g., Jin et al. [18]), we adopt a similar solution idea as in [10] to attach to this class of problems an upper limit on the funding level of the terminal wealth. In the continuous-time mean-LPM and mean-CVaR portfolio optimization models, if the terminal wealth is unlimited, the investor will act extremely aggressively to push his terminal wealth to the infinity. Adding a limit on the funding level will tame such an irregular portfolio policy to a reasonable level. Thus, such an upper bound can be also regarded as a designing variable to control the aggressiveness level of the investor. We further prove that the probability that the terminal wealth reaches such an upper bound is decreasing with respect to the magnitude of the upper level. For general market opportunity set, we prove the exitance and uniqueness of Lagrangian multipliers, which is the key step to apply the martingale approach. These theoretical results pave a foundation to develop numerical solution schemes to solve dynamic mean-LPM and mean-CVaR portfolio optimization problems. When the market opportunity set is deterministic, we further derive semi-analytical portfolio policies for both the mean-LMP and mean-CVaR portfolio optimization problems. The dynamic mean-LPM portfolio policy demonstrates very distinct features when compared with the dynamic MV portfolio policy. When the market condition is good, the mean-LPM investor tends to invest more aggressively in the risky assets when compared to an MV investor. When the market condition is in the medium state, the mean-LPM investor prefers to allocate more wealth in the risk-free asset. However, when the market condition is in a bad state, the mean-LPM investor allocates again more wealth in the risky assets than the MV investor. This phenomena can be regarded as the gambling effect of dynamic mean-LPM investors. In summary, the mean-LPM investment policies show a feature of a two-side threshold type, i.e., at any time , when the current wealth is, respectively, below or above certain levels, the investor increases his allocations in the risky assets. As for the dynamic mean-CVaR portfolio policy, our experiment result with real market data shows that the CVaR measure can be improved significantly when compared with the buy-and-hold mean-CVaR portfolio policy of a static type.

The remaining of the paper is organized as follows. We present the market setting and the dynamic mean-LMP and dynamic mean-CVaR portfolio optimization problem formulations in Section LABEL:se_formulation. We derive the optimal portfolio policies for the dynamic mean-LPM and dynamic mean-CVaR optimization problems in Section LABEL:se_LPM and LABEL:se_cvar, respectively. We then present illustrative examples to compare the dynamic mean-LPM portfolio policy with the dynamic mean-variance portfolio policy and the dynamic mean-CVaR portfolio policy with the static portfolio policy in Section LABEL:se_example. Finally, we conclude our paper in Section LABEL:Conclusion. Throughout the entire paper, notation denotes the indicator function, i.e., if the condition holds true and , otherwise; denotes the transpose of matrix , and denotes the nonnegative part of , i.e., . To simplify our notations, we use for which means the -th power function of . Finally, the cumulative distribution of the standard normal random variable is denoted by .

2 Market Setting and Problem Formulations

We consider a market with risky assets and one risk free asset which can be traded continuously within time horizon . All the randomness are modeled by a complete filtrated probability space , on which an adapted -dimensional Brownian motion is defined, where and are mutually independent for all . Let be the set of -valued, -adapted and square integrable stochastic processes, and the set of valued -measurable random variables.

The price process of the risk-free asset is governed by the following ordinary differential equation,

\hb@xt@.01(2.1)

where is the risk free return rate, which is measurable scalar-valued stochastic process. The price process of the risky assets satisfies the following system of stochastic differential equations (SDE):

\hb@xt@.01(2.2)

where and are the appreciation rate and volatility, respectively. We assume that all and are uniformly bounded, scalar-valued -measurable stochastic processes. Furthermore, we assume that the volatility matrix satisfies the following nondegeneracy condition,

\hb@xt@.01(2.3)

for some 111‘a.s.’ stands for ‘almost surely’, which excludes events with zero occurrence probability. In the following discussion, we simply ignore such a term for the random variables that satisfy certain condition.. Under the above setting, we have a complete market model for the securities.

An investor with initial wealth enters the market at time and continuously allocates his wealth in the risky assets and the risk-free asset within time horizon . Let be the total wealth of the investor at time . Denote the portfolio process by with , where is the dollar amount allocated to risky asset at time . As we do not consider in this research the transaction cost during the investment process, the wealth process of the investor, , then satisfies the following stochastic differential equation (SDE),

\hb@xt@.01(2.4)

where is the excess return defined by

In this research, we focus our investigation on mean-downside risk portfolio optimization. In particularly, we are interested in studying the following mean-LPM model,

Subject to

where is the minimum expected wealth which the investor would like to attain, is an upper bound of the attainable final wealth imposed by the investor, is a given benchmark level, and is a given nonnegative integer, which represents the order of the moment. Adopting model implies that the investor only cares about the scenarios where is less than the benchmark level , which the investor sets as a threshold for “disastrous” terminal wealth. When = 0, from our notations, we have and thus , which is the disaster probability considered by Roy in his pioneering safety-first principle [32], while can be viewed as the disaster level. When = 1 and = , the downside risk measure becomes the semi-absolute deviation (or the target semi-absolute-deviation). When = 2 and = , the downside risk measure yields the semi-variance (or the target semi-variance). Let be a given safe-level of the terminal wealth. One possible candidate of could be

\hb@xt@.01(2.5)

which is the expected terminal wealth when investing all initial wealth in the risk free account. For the upper bound , we reasonably assume .

In our work, we also study dynamic mean-CVaR portfolio optimization. We define first the loss of investment as follows,

\hb@xt@.01(2.6)

We adopt the definition of CVaR by Rockafellar and Uryasev [31] for investment loss and use the notation to denote the CVaR of the investment loss. The mean-CVaR portfolio optimization model is now formally posted as follows,

Subject to

where all the other notations are defined the same as in .

As we will demonstrate later in this paper, the upper bound, , imposed on the terminal wealth essentially controls the aggressiveness of the portfolio policy. The larger the value of , the more aggressive the portfolio policy becomes. If we let go to infinite, both problems and will become ill-posed (see, e.g., [18]), i.e., the investor would take an infinite position. From the view point of real applications, any portfolio that generates extremely high level of terminal wealth is not realistic. Thus, imposing an upper bound on the terminal wealth, as proposed in [10], is reasonable and justifiable. Furthermore, such an upper bound can be also regarded as a designing variable to control the aggressiveness level of the investor. We also prove that the probability that the terminal wealth reaches its upper bound is monotonically decreasing with respect to the level of the upper bound. Thus, a formulation with a very large upper bound can be regarded as an approximation to the formulation without an upper bound. Note also that the no-bankruptcy constraint at the terminal time, , actually ensures no-bankruptcy for the entire wealth process, i.e., , for (see Proposition 2.1 in [7]).

3 Optimal Portfolio Policy For Dynamic Mean-LPM Formulation

We develop in this section a solution scheme for problem using the martingale approach (see, for examples, [28] and [20]). The main idea of the martingale approach is to find first the optimal terminal wealth by solving a static optimization problem and to identify then the optimal portfolio policy process to replicate (generate) such an optimal wealth distribution of .

3.1 Optimal terminal wealth

From our complete market setting in (LABEL:def_risky_sde), we can find a unique equivalent martingale measure (EMM) such that the discounted price processes of the risk assets are martingale. Let the Radon-Nikodým derivative of the EMM, , with respect to the original measure be , i.e., , where is an -measurable random variable. From the Girsanov Theorem [20], the Radon-Nikodým derivative process can be expressed as the exponential martingale, , where is vector-valued -adapted stochastic process vector such that the choice of makes the process to be the Brownian motion under probability . To eliminate the drift term of the discount price process of the securities, we let be

\hb@xt@.01(3.1)

Then, we define the state price density as which satisfies the following SDE,

\hb@xt@.01(3.2)

or, equivalently, we can express as

In the literature, is also referred as the deflator process, which transfers the discounted wealth process to a martingale, i.e., we have

for any . By using such a property, the optimal terminal wealth of the problem can be found by solving the following static optimization problem,

Subject to \hb@xt@.01(3.3)
\hb@xt@.01(3.4)

Before we solve problem () we need the following lemma.

Lemma 1

Given the following two problems () and (),

Subject to:

and

where is a convex set, is a scalar-valued convex function, is a random variable, , , and . If solves problem for some and and satisfies and , then solves problem with and . On the other hand, if problem has a solution , then there exist and such that also solves problem .

We place the proof of Lemma LABEL:lem_lag in the Appendix. Lemma LABEL:lem_lag basically shows that problem can be solved by investigating its corresponding Lagrange relaxation problem (). Before we give the main results, we define the following set,

\hb@xt@.01(3.5)

Considering the convexity issue of function , we separate the cases with from the ones with in problem . The optimal terminal wealth of problem () is given by the following two theorems separately for these two situations.

Theorem 1

When , the optimal solution of problem () takes one of the following two forms. (i) If , the optimal solution can be expressed as

\hb@xt@.01(3.6)

where the Lagrange multipliers and satisfy the following conditions,

\hb@xt@.01(3.7)
\hb@xt@.01(3.8)

and if inequality (LABEL:thm_lpm_q>1_eq1) holds strictly, . (ii) If , then any random variable is optimal for problem .

Proof. To simplify the notation, we use and for and , respectively, in the following discussion. Introducing Lagrange multipliers and , respectively, for constraints (LABEL:A_lpm_cnst1) and (LABEL:A_lpm_cnst2) in problem () yields the following Lagrange relaxation of problem ,

\hb@xt@.01(3.9)

Ignoring the constant terms, we solve first the inner point-wise optimization problem,

\hb@xt@.01(3.10)

We first assume that . Since is a random variable, the optimal solution of problem (LABEL:lpm_g) depends on different values of . If , problem (LABEL:lpm_g) reduces to

\hb@xt@.01(3.11)

It can be verified that is convex with respect to in the range . The stationary point of function satisfies

If , then we have . Thus, the optimal solution of problem (LABEL:lpm_g(i)) is with . If , then is a monotonically decreasing function with respect to , which implies that the optimal solution of problem (LABEL:lpm_g(i)) is with . If , the optimal solution is with . If , problem (LABEL:lpm_g) becomes

\hb@xt@.01(3.12)

The optimal solution is with if and with if . As a summary, we can conclude that, when , the optimal solution of problem (LABEL:lpm_g0) is

\hb@xt@.01(3.13)

Due to Lemma LABEL:lem_lag, we know that the Lagrange method provides the necessary and sufficient condition of problem . Thus, solves problem () when it satisfies the conditions given in (LABEL:thm_lpm_q>1_eq1) and (LABEL:thm_lpm_q>1_eq2) for case (i).

If or and , we know that is a strictly decreasing function with respect . Thus, the optimal solution of problem (LABEL:lpm_g0) is , which never satisfies the constraint (LABEL:A_lpm_cnst2). The only remaining case is and . Under this situation, the relaxation problem (LABEL:lpm_g) degenerates to . That is to say, any that satisfies is the optimal solution of problem (LABEL:lpm_g0). Thus, is the solution of problem when also satisfies (LABEL:A_lpm_cnst1) and (LABEL:A_lpm_cnst2). Equivalently, we can use the set as in (LABEL:def_X) to characterize such a solution. We thus complete the proof for case (ii).     

Theorem 2

When , the following holds true for problem (): (i) If , the following solution solves problem problem (),

\hb@xt@.01(3.14)

where and satisfy the following conditions,

\hb@xt@.01(3.15)
\hb@xt@.01(3.16)

In addition, if inequality (LABEL:thm_lpm_q<1_eq1) holds strictly, then . (ii) If where is defined in (LABEL:def_X), then any random variable is optimal to problem . (iii) When , if problem () has optimal solution, the solution can only take one of the forms in case (i) or (ii).

Proof. We use the notations similar to the ones in the proof of Theorem LABEL:thm_lpm_q>1. Fisrt we assume . When , problem (LABEL:lpm_g(i)) is to minimize a concave function with respect to . Thus, the minimizer of (LABEL:lpm_g(i)) can only be at the boundary points, either with or with , respectively. Comparing the function values of the two boundary points yields if and if . Combined with the case where in problem (LABEL:lpm_g), we have the solution of the Lagrange relaxation problem (LABEL:lpm_g0) as follows,

From Lemma LABEL:lem_lag, we know that is the optimal solution of problem if satisfies (LABEL:thm_lpm_q<1_eq1) and (LABEL:thm_lpm_q<1_eq2) given in case (i). The proof of case (ii) is the same as the proof of case (ii) in Theorem LABEL:thm_lpm_q>1, which is correspondent to the situation and . When , the objective function of problem is convex. From Lemma LABEL:lem_lag, the Lagrange method characterizes all the solutions of .     

From Theorems LABEL:thm_lpm_q>1 and LABEL:thm_lpm_q<1, we know that the optimal terminal wealth of problem depends on different values of Lagrange multipliers and , which in turn depend on the parameters of problem (), e.g., the target and benchmark . We will discuss in details the relationship between these parameters and the Lagrange multipliers in Section LABEL:sse_Lagrange.

Theoretically speaking, once the optimal terminal wealth is known, the optimal portfolio policy can be characterized by solving the following backward stochastic differential equation(BSDE),

\hb@xt@.01(3.17)

Let and be the solution of the linear BSDE in (LABEL:BSDE). Then the optimal portfolio policy satisfies . In a general setting, there could be no explicit solution for the BSDE in (LABEL:BSDE). However, when the market parameters are deterministic, the optimal wealth process and portfolio policy can be expressed explicitly, as will be shown in Section LABEL:sse_determin. From the BSDE in (LABEL:BSDE) we can get the upper and lower bounds of the whole wealth process. Recall that a no-bankruptcy constraint at the terminal time, , actually ensures no-bankruptcy for the entire wealth process, i.e., , for (see [7]). Similarly, we will show now that the upper bound imposed on the terminal wealth also induces an upper bound on the entire wealth process.

Proposistion 1

In problem (), if we have , where , then the wealth process is bounded by

\hb@xt@.01(3.18)

where process is defined in (LABEL:def_z).

Proof. Let us consider the following BSDE with boundary condition of ,

\hb@xt@.01(3.19)

According to [21], the solution of (LABEL:prop_bound_bsde) can be expressed as

\hb@xt@.01(3.20)

By using the comparison theorem (Theorem 2.2 in [21]), we can conclude that for . We can use the similar argument for the lower bound of .     

3.2 The existence of Lagrange multipliers

From Theorems LABEL:thm_lpm_q>1 and LABEL:thm_lpm_q<1, we know that the Lagrange multipliers and for problem () can be determined by checking the conditions in (LABEL:thm_lpm_q>1_eq1) and (LABEL:thm_lpm_q>1_eq2) for ; or the conditions in (LABEL:thm_lpm_q<1_eq1) and (LABEL:thm_lpm_q<1_eq2) for , respectively. However, at this point, we cannot guarantee the existence and uniqueness of these Lagrange multiplies. Furthermore, it is unclear at this stage under which condition the optimal solution of problem () takes the form (i) or (ii) in both Theorems LABEL:thm_lpm_q>1 and LABEL:thm_lpm_q<1. As pointed out in [19], the existence of the Lagrange multipliers is related to the well-poseness of the problem itself. On the other hand, from the application viewpoint, investors often adjust the investment target to generate different efficient portfolios for comparison. Thus, it is important to investigate the impact of the parameters and on problem . Before we state the main result, we need the following assumption.

Assumption 1

The probability density function of , , is a continuous function.

We first define, for some positive number , some functions of -th order partial moments with respect to random variable ,

Obviously, when , reduces to the distribution function of . From the definition of , it is evident that is a monotonically increasing function with respect to , for . Under Assumption LABEL:asmp_pdf, we can also show that and are also monotonically increasing functions for . Essentially, taking the derivative of and with respect to gives rise to

which imply the monotonicity of and for .

For a given problem (), we define the following parameters and , which play key roles in solving problem ,

\hb@xt@.01(3.21)
\hb@xt@.01(3.22)

Note that and are well defined. Due to the monotonicity of , letting and yields and , respectively. The condition implies the existence of . The similar argument also applies to and .

The following propositions ensure the existence and uniqueness of the Lagrange multipliers and in Theorems LABEL:thm_lpm_q>1 and LABEL:thm_lpm_q<1.

Proposistion 2

For problem with , under Assumption LABEL:asmp_pdf, the following results hold.

  • If , the solution of problem () is given by (LABEL:thm_lpm_q>1_X) and there is a unique pair of and satisfying the conditions in (LABEL:thm_lpm_q>1_eq1) and (LABEL:thm_lpm_q>1_eq2) with equality holding for (LABEL:thm_lpm_q>1_eq1).

  • If and , the solution of problem () is given by (LABEL:thm_lpm_q>1_X) with and satisfying the conditions in (LABEL:thm_lpm_q>1_eq1) and (LABEL:thm_lpm_q>1_eq2).

  • If and , then problem has multiple solutions which are characterized by set defined in (LABEL:def_X) and one of such solutions is

    \hb@xt@.01(3.23)

Proof. (i) We prove this result by identifying the range of under which the equality holds for both (LABEL:thm_lpm_q>1_eq1) and (LABEL:thm_lpm_q>1_eq2). To simplify our notation, we change the variables and in conditions (LABEL:thm_lpm_q>1_eq1) and (LABEL:thm_lpm_q>1_eq2) to

\hb@xt@.01(3.24)

respectively. Note that and . In the following part, we let . When both equalities hold, the conditions in (LABEL:thm_lpm_q>1_eq1) and (LABEL:thm_lpm_q>1_eq2) become a system of two equations,

\hb@xt@.01(3.25)
\hb@xt@.01(3.26)

where

We show that and are monotonically increasing functions with respect to both and . To simplify the notations, we do not write out the arguments in and explicitly, unless necessary. Assumption LABEL:asmp_pdf implies the differentiability of and . Taking the derivatives of and with respect to and , respectively, yields the following results for and ,

\hb@xt@.01(3.27)

which imply the monotonicities of the and with respect to both and . Thus, we can determine the ranges of and as and , respectively. To solve the system of equations (LABEL:prop_lagq>1_eq1) and (LABEL:prop_lagq>1_eq2), we define the following function as follows,

For a given , due to the monotonicity of , holds true. Thus, there exists a unique that satisfies . Thus, is well defined. We will show that is a monotonically decreasing function with respect to . Since holds, we have

\hb@xt@.01(3.28)

Checking the derivative of with respect to gives rise to

\hb@xt@.01(3.29)

where the last equality is based on (LABEL:prop_lagq>1_delta_rho). Since , the sign of depends on the numerator of (LABEL:prop_lagq>1_dI3). Combining (LABEL:prop_lagq>1_eq0) with (LABEL:prop_lagq>1_dI3) further yields the following,

\hb@xt@.01(3.30)

where the last inequality is from the Cauchy Schwarz Inequality,

Inequality (LABEL:prop_lagq>1_eq3) implies that , which proves the monotonicity of . Thus, for any , if is in the range space of , we can always find a unique such that . Now, we only have to fix the range of . Due to the definition of , we define

\hb@xt@.01(3.31)
\hb@xt@.01(3.32)

From (LABEL:prop_lagq>1_eq0) and (LABEL:prop_lagq>1_delta_rho), we know that , if holds. It is not hard to see . We can find the corresponding when as follows,

Taking and yields the upper limit of ,

\hb@xt@.01(3.33)

Now, we focus on the lower limit of . Since when holds, there are two candidates of , namely, or the corresponding when . If , we find by checking

which leads to the lower limit of ,

\hb@xt@.01(3.34)

If