Loading [MathJax]/jax/output/SVG/jax.js
Review Topical Sections

Endoplasmic reticulum, oxidative stress and their complex crosstalk in neurodegeneration: proteostasis, signaling pathways and molecular chaperones

  • Received: 03 August 2017 Accepted: 09 October 2017 Published: 20 September 2017
  • Cellular stress caused by protein misfolding, aggregation and redox imbalance is typical of neurodegenerative disorders such as Parkinson’s disease (PD) and Amyotrophic Lateral Sclerosis (ALS). Activation of quality control systems, including endoplasmic reticulum (ER)-mediated degradation, and reactive oxygen species (ROS) production are initially aimed at restoring homeostasis and preserving cell viability. However, persistent damage to macromolecules causes chronic cellular stress which triggers more extreme responses such as the unfolded protein response (UPR) and non-reversible oxidation of cellular components, eventually leading to inflammation and apoptosis. Cell fate depends on the intensity and duration of stress responses converging on the activation of transcription factors involved in the expression of antioxidant, autophagic and lysosome-related genes, such as erythroid-derived 2-related factor 2 (Nrf2) and transcription factor EB respectively. In addition, downstream signaling pathways controlling metabolism, cell survival and inflammatory processes, like mitogen activated protein kinase and nuclear factor-kB, have a key impact on the overall outcome.
    Molecular chaperones and ER stress modulators play a critical role in protein folding, in the attenuation of UPR and preservation of mitochondrial and lysosomal activity. Therefore, the use of chaperone molecules is an attractive field of investigation for the development of novel therapeutic strategies and disease-modifying drugs in the context of neurodegenerative diseases such as PD and ALS.

    Citation: Giulia Ambrosi, Pamela Milani. Endoplasmic reticulum, oxidative stress and their complex crosstalk in neurodegeneration: proteostasis, signaling pathways and molecular chaperones[J]. AIMS Molecular Science, 2017, 4(4): 424-444. doi: 10.3934/molsci.2017.4.424

    Related Papers:

    [1] Lin Fan, Shunchu Li, Dongfeng Shao, Xueqian Fu, Pan Liu, Qinmin Gui . Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations. AIMS Mathematics, 2022, 7(7): 11972-11991. doi: 10.3934/math.2022667
    [2] Cheng Chen . Hyperbolic function solutions of time-fractional Kadomtsev-Petviashvili equation with variable-coefficients. AIMS Mathematics, 2022, 7(6): 10378-10386. doi: 10.3934/math.2022578
    [3] Numan Yalçın, Mutlu Dedeturk . Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method. AIMS Mathematics, 2021, 6(4): 3393-3409. doi: 10.3934/math.2021203
    [4] Raed Qahiti, Naher Mohammed A. Alsafri, Hamad Zogan, Abdullah A. Faqihi . Kink soliton solution of integrable Kairat-X equation via two integration algorithms. AIMS Mathematics, 2024, 9(11): 30153-30173. doi: 10.3934/math.20241456
    [5] Mohammed Alrehili . Managing heat transfer effectiveness in a Darcy medium with a vertically non-linear stretching surface through the flow of an electrically conductive non-Newtonian nanofluid. AIMS Mathematics, 2024, 9(4): 9195-9210. doi: 10.3934/math.2024448
    [6] J.-C. Cortés, A. Navarro-Quiles, J.-V. Romero, M.-D. Roselló . First-order linear differential equations whose data are complex random variables: Probabilistic solution and stability analysis via densities. AIMS Mathematics, 2022, 7(1): 1486-1506. doi: 10.3934/math.2022088
    [7] Mustafa Inc, Hadi Rezazadeh, Javad Vahidi, Mostafa Eslami, Mehmet Ali Akinlar, Muhammad Nasir Ali, Yu-Ming Chu . New solitary wave solutions for the conformable Klein-Gordon equation with quantic nonlinearity. AIMS Mathematics, 2020, 5(6): 6972-6984. doi: 10.3934/math.2020447
    [8] Andrey Muravnik . Nonclassical dynamical behavior of solutions of partial differential-difference equations. AIMS Mathematics, 2025, 10(1): 1842-1858. doi: 10.3934/math.2025085
    [9] Ziying Qi, Lianzhong Li . Lie symmetry analysis, conservation laws and diverse solutions of a new extended (2+1)-dimensional Ito equation. AIMS Mathematics, 2023, 8(12): 29797-29816. doi: 10.3934/math.20231524
    [10] M. Hafiz Uddin, M. Ali Akbar, Md. Ashrafuzzaman Khan, Md. Abdul Haque . New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative. AIMS Mathematics, 2019, 4(2): 199-214. doi: 10.3934/math.2019.2.199
  • Cellular stress caused by protein misfolding, aggregation and redox imbalance is typical of neurodegenerative disorders such as Parkinson’s disease (PD) and Amyotrophic Lateral Sclerosis (ALS). Activation of quality control systems, including endoplasmic reticulum (ER)-mediated degradation, and reactive oxygen species (ROS) production are initially aimed at restoring homeostasis and preserving cell viability. However, persistent damage to macromolecules causes chronic cellular stress which triggers more extreme responses such as the unfolded protein response (UPR) and non-reversible oxidation of cellular components, eventually leading to inflammation and apoptosis. Cell fate depends on the intensity and duration of stress responses converging on the activation of transcription factors involved in the expression of antioxidant, autophagic and lysosome-related genes, such as erythroid-derived 2-related factor 2 (Nrf2) and transcription factor EB respectively. In addition, downstream signaling pathways controlling metabolism, cell survival and inflammatory processes, like mitogen activated protein kinase and nuclear factor-kB, have a key impact on the overall outcome.
    Molecular chaperones and ER stress modulators play a critical role in protein folding, in the attenuation of UPR and preservation of mitochondrial and lysosomal activity. Therefore, the use of chaperone molecules is an attractive field of investigation for the development of novel therapeutic strategies and disease-modifying drugs in the context of neurodegenerative diseases such as PD and ALS.


    The effectiveness of financial activities depends on the choice of investment decision-making. Recent research has increasingly focused on optimal investment strategy selection in finance and operations. Key factors drive effective solutions in this area include selecting an appropriate risk measure to capture potential extreme losses, assessing how market fluctuations impact investments, and balancing returns against risks. In classical Markowitz mean-variance models, risk is measured by deviations from the mean, such as variance or standard deviation. However, relying on variance as a risk measure presents serious limitations. Compared to variance, measuring the downside risk of a portfolio is more critical, an insight long recognized by both scholars and practitioners.

    Until the late 1980s, the Basel Committee emphasized the importance of widely accepted risk standards and recommended introducing quantitative models based on mathematical and statistical principles. Value-at-Risk (VaR) was explicitly recommended for evaluating the capital adequacy and market risk of commercial banks for the first time. However, VaR has notable limitations, particularly its inability to adequately measure tail risk. When actual losses exceed the VaR threshold, their magnitude and acceptability remain unknown. To address these shortcomings, recent research has focused on coherent risk measures. Artzner et al. [1] were among the first to investigate this issue, introducing the concept of coherent risk measures. Following their work, research on coherent risk measures has attracted significant attention, leading to significant advancements, including Expected Shortfall (ES) [2], Conditional Value-at-Risk (CVaR) [3], spectral risk measures [4], and one-sided moment measures [5]. A substantial body of literature has explored methods for minimizing these coherent risk measures, with CVaR emerging as the most extensively studied and widely applied. Rockafellar and Uryasev demonstrated that solving a simple convex optimization problem enables the simultaneous computation of both CVaR and VaR for a portfolio. CVaR offers an efficient approach for solving portfolio optimization problems, facilitating large-scale computations that would otherwise be infeasible.

    Corresponding to the research on risk measures, the practical applications of portfolio selection models have also expanded. Private and institutional investors are developing dynamic techniques and tools to improve security price predicting and enhance investment capital management. Numerous portfolio selection models have been proposed, employing diverse solution techniques and applications across different markets. However, a significant limitation in existing research is the assumption that the distribution of risk asset prices or returns is known in advance or fully specified. This assumption often renders many risk management methods and optimal investment strategy models impractical for real-world investment decision-making, as precise characterizations of security returns are often unavailable. In response, modern optimization methods for decision-making under uncertainty, such as robust optimization techniques, have gained prominence in risk management and portfolio selection. Recently, various robust risk measures and corresponding robust portfolio selection models have emerged, yielding several valuable results. Nonetheless, many issues remain unresolved or require further refinement. This paper, therefore, focuses on constructing and solving robust portfolio selection models within this framework.

    Robust optimization has emerged as a powerful tool for addressing optimization problems under uncertainty. Soyster [6] initially introduced the method of robust optimization, and definitions for robust feasible solutions and optimal solutions were later provided by Ben-Tal and Nemirovski [7] and Ghaoui [8]. Garlappi et al. [9] addressed the mean-variance robust portfolio selection problem, assuming only the mean is uncertain and belonging to a box uncertainty set. Costa and Paiva [10], Goldfarb and Iyengar [11], and Lu [12] investigated robust portfolio selection within the mean-variance framework. Goldfarb and Iyengar [11] considered a factor model for asset stochastic returns, constructing uncertainty sets for the model parameters using statistical processes. Lu [12] studied the robust portfolio selection problem using a joint ellipsoidal uncertainty set to describe the model parameters, demonstrating that the problem can be reformulated as a cone programming problem. Halldórsson and Tütüncü [13] extended these results [10,11,12] by applying interior point algorithms to address the robust mean-variance portfolio selection problem with mean vectors and covariance matrix parameters modeled as box uncertainty sets. Popescu [14] examined the robust mean-variance (M-V) portfolio selection problem when the moment information of a given return vector distribution is known. They demonstrated that, for a broad class of objective functions, finding a robust solution is equivalent to solving a parameterized quadratic program. Natarajan et al. [15] focused on the worst-case CVaR robust portfolio selection model when only partial moment information of the stochastic return variables is known. Zhu and Fukushima [16] introduced a different type of uncertainty, where instead of focusing on the first and second moments of the portfolio, the uncertainty is in the distribution of portfolio returns themselves. Distributionally robust optimization addresses the uncertainty in asset return distributions by considering a set of possible distributions rather than relying on a single estimated distribution. This approach is beneficial when the underlying distribution is unknown or subject to change. The Wasserstein metric is commonly employed to define ambiguity sets, allowing for a robust optimization framework adaptable to various scenarios [17,18]. Subsequently, Huang et al. [19] applied the methods from [16] to portfolio selection problems with uncertain termination times. In practice, there are various methods to handle uncertainty in the covariance matrix of a model. Some approaches involve additional factors in the return model [20], while others consider confidence intervals for individual covariance matrices [14]. Even when the uncertainty set is defined simply as a collection of possible scenarios for the covariance matrix, the advantages of such approaches are well recognized [10,21]. Best and Grauer [22] and Black and Litterman [23] studied the sensitivity of optimal portfolio estimates to uncertainty in average returns.

    Dynamic risk measures play a crucial role in assessing the risk of financial portfolios over time. Large portfolios that use the CVaR measure often exhibit non-smooth characteristics. To address this, [24] proposed a derivative-free method for nonsmooth functions. Regime-switching models account for the nonstationarity of financial markets by allowing parameters to shift across different regimes. These models can capture phenomena such as volatility clustering and fat tails frequently observed in financial data. For instance, the Markov regime-switching GARCH model has been widely used to model asset returns under varying market conditions [25]. [26] introduced explicit CRRA equilibrium strategies for two-player stochastic investment games under Markovian regime switching, while [27] derived globally optimal solutions for incomplete regime-switching markets. [28] proposed a novel VIX-based candlestick predictor with market regime analysis. Quantum-inspired optimization [29] and robust genetic strategies [30] provide scalable frameworks for high-dimensional and dynamic challenges, aligning with regime-switching CVaR portfolios. Additionally, AI techniques for dynamic risk management have also been applied across various fields [31].

    In financial portfolio management, optimizing asset allocation while dynamically managing risk remains a critical challenge. Traditional robust risk measurement and portfolio selection models often rely on worst-case scenarios, resulting in overly conservative investment decisions that fail to accurately reflect the impact of market changes on the uncertainty set. Existing studies on robust portfolio selection models with known matrix uncertainty either focus on single-period scenarios or lack analytical solutions. However, the real financial market is highly dynamic, particularly in medium and long-term investments, and multi-period risk models are essential for effectively managing risks over investment horizons. Single-period risk models have limitations in offering optimal long-term investment strategies. Therefore, extending single-period risk measures to a multi-period framework is highly significant. Furthermore, most portfolio selection models assume deterministic information, such as known distribution functions, but in practice, market parameters are inherently uncertain. Even minor parameter changes can significantly affect investment outcomes, potentially leading to suboptimal or infeasible solutions. Robust optimization provides a powerful method to address parameter uncertainty. Given these challenges, this study aims to bridge these gaps by developing a multi-period robust portfolio selection model that effectively integrates dynamic uncertainty considerations with robust optimization techniques.

    This study focuses on a market consisting of multiple risky assets and one risk-free asset, extending previous research [16] by including the risk-free asset. The returns of the risky assets are characterized by a given mean vector and covariance matrix, forming an uncertainty set distinct from the Wasserstein ambiguity set [18] and the asymmetric distribution uncertainty set [32]. We consider a multi-period robust portfolio selection model that utilizes robust CVaR as the risk measure, contrasting it with the mean lower partial moment [32], and solve the problem via dynamic programming, which differs from the SOCP optimization approach [33].

    The main contributions of the paper include:

    ● The proposed multi-period investment strategy is formulated from a dynamic perspective, allowing investors to adjust their strategies according to market conditions throughout the holding period to enhance returns.

    ● We utilize regime-switching techniques to capture the dynamic dependencies between consecutive periods, adjusting the uncertainty set based on the first and second moments to reflect these dynamic relationships.

    ● The constructed uncertainty set features a mean vector that follows a Markov process. We demonstrate that the optimal investment strategy, derived recursively, depends on this mean vector, ensuring that the optimal strategy adapts to the state of the uncertainty set. This dynamic investment strategy offers a more realistic alternative to static strategies.

    ● By using wealth dynamic equations as constraints and leveraging existing solution techniques, we derive an analytical optimal investment strategy based on dynamic programming principles.

    This paper is structured as follows. Section 2 introduces the multi-stage robust portfolio selection model, incorporating a regime-switching technique to capture the dynamic correlations. Section 3 presents an approach to derive the analytical optimal solution using dynamic programming. Section 4 details our proposed method, which recursively breaks down the problem from the current stage back to the initial stage. Section 5 provides numerical analysis and results. Finally, Section 6 concludes the paper.

    We consider a security market consisting of n risky assets and one risk-free asset with return R. To maintain model tractability, we use CVaR as the fundamental risk measure when constructing the multi-period robust portfolio selection model.

    To characterize the dynamic changes in the stochastic returns of risky assets, we define a probability space (Ω,F,P). The sigma algebra Fk represents all available information up to time k, with the assumption that F0={Ω,} and FK=F, where K denotes the total investment period. Thus, since F0F1FK, the collection {Fk} forms a filtration. At the beginning of each period, the current wealth is reallocated among all assets. We denote the proportion of wealth allocated to n risky assets at stage k by the vector xk=(x1k,,xnk)T.

    Let ξk=(ξ1k,,ξnk)T represent the vector of random returns for the n risky assets at time k. This vector is defined as a random variable on the probability space (Ω,Fk,P) for k=1,2,,K. Moreover, for each k, ξk is Fk-measurable, indicating that the stochastic process {ξk,k=1,2,,K} is adapted to the filtration {Fk,k=1,2,,K}.

    We assume that the first and second moments of asset returns are known. Due to dynamic dependence, the mean vector at time k is conditionally dependent on the information available at time k1, represented as μk=EPk1[ξk]. To capture the dynamic correlations of return rates, we employ a regime-switching model. In this framework, the regime process follows a Markov chain, where the set of possible regimes is constructed by m regimes U={μ1,μ2,,μm}. The transition probability from regime μi at time k to regime μj at time k+1 is denoted as Pμiμj(k,k+1)=P{μk+1=μj|μk=μi}. We assume the Markov chain is time-homogeneous with stationary transition probabilities. Therefore, the transition probability matrix at time k is denoted as

    Pk=(Pμ1μ1Pμ1μ2Pμ1μmPμ2μ1Pμ2μ2Pμ2μmPμmμ1Pμmμ2Pμmμm).

    Consequently, the state of μk at time k+1 depends only on its state at time k, satisfying the Markov property. This implies that, throughout the investment process, the mean for the next period relies on the mean return of the previous period. Investors can adjust their strategies at each stage of the investment horizon in response to market fluctuations, thereby optimizing returns.

    Furthermore, we define the covariance matrix of risky asset returns at stage k as Γk=Cov[ξk]. For simplicity, we follow the approach in [15] and assume that Γk=Γ0, k=1,,K across different periods of the investment horizon, where Γ0 denotes that Γ is positive definite.

    Next, we describe the uncertainty in the distribution of return rates, assuming that ξk at stage k belongs to a given uncertainty set Dk defined by its first two moments,

    Dk={πkE[ξk]=μk,Cov[ξk]=Γ}.

    Suppose an investor joins the market at time 0 with an initial wealth of w0=1. The investor plans to allocate this wealth in the securities market over K periods, where the cumulative return rate at stage k is

    rk=wkw0w0,k=1,,K.

    We define r_k as the minimum required cumulative return rate for each period, ensuring r_KrK1R for k=1,,K. We assume that the investment process is self-financing, leading to the dynamic equation rk1+E[ξk]Txk+R(1xTke)=r_k,k=1,,K, where e=[1,,1]T. Without constraints, excessive leverage or short positions in risky assets may arise. However, the inclusion of a risk-free asset along with a minimum return constraint addresses this issue, resulting in a more stable portfolio. Furthermore, the presence of a risk-free asset allows investors to allocate capital between risky and risk-free assets, enabling dynamic position adjustments under different market regimes. Throughout the investment process, the investor constantly reallocates their wealth among n risky assets and one risk-free asset at the beginning of each period. The terminal total wealth at the end of stage k is denoted by wk, while wk can be viewed as the potential loss at stage k.

    The optimal investment strategy must be determined at the decision-making outset for investors. When selecting x1, the actual return rate ξ1 is unknown. Similarly, when formulating the investment strategy xk(k2), the return ξk remains uncertain. Therefore, xk should depend on ξk1 rather than ξk, making xk a variable influenced by uncertain data ξk1. This implies that the investment decision xk+1 for stage k+1 is made based on the information from the previous period, without knowledge of the current return ξk+1, for k=1,,K.

    Investors usually aim to minimize risk while maximizing terminal wealth in multi-stage investment scenarios. We continue to use CVaR as the fundamental risk measure. Based on the aforementioned uncertainty set, we derive a robust CVaR measure to control total risk from any intermediate moment until the end of the investment period.

    We define the loss function at time k as follows:

    f(xk,ξk)=(xTkξk+R(1xTke)).

    Rockafellar and Uryasev [3,34] demonstrated that CVaR can be computed by minimizing the auxiliary function

    Fβk(xk,αk)=αk+11βkE[(f(xk,ξk)αk)+],

    where αk represents the threshold for the loss function, and βk(0,1) denotes the confidence level. Then,

    CVaRβk(xk)=minαkRFβk(xk,αk).

    We express the robust CVaR risk measure as follows:

    RCVaRβk(xk)=maxπkDkCVaRβk(xk)=maxπkDkminαkRFβk(xk,αk).

    Unlike existing literature, our approach ensures that investors minimize total risk while ensuring that the return rates do not drop below a pre-specified threshold in each period. Let Vk(wk1) represent the optimal target value at stage k. Under these stochastic market conditions, we formulate the multi-stage robust portfolio selection model using dynamic programming principles:

    VK(rK1)=minxKRnmaxξKDK(μK,Γ)minαKR{αK+11βKE[(RαKxTK(ξKRe))+]} s.t. rK1+(E[ξK])TxK+R(1xTKe)=r_K;VK1(rK2)=minxK1maxξK1DK1(μK1,Γ)minαK1{αK1+11βK1E[(RαK1xTK1(ξK1Re))+]+E[VK(rK1)]} s.t. rK2+(E[ξK1])TxK1+R(1xTK1e)=r_K1;V1(1)=minx1maxξ1D1(μ1,Γ)minα1{α1+11β1E[(Rα1xT1(ξ1Re))+]+E[V2(r1)]} s.t. (E[ξ1])Tx1+R(1xT1e)=r_1. (2.1)

    It is important to emphasize that when making decisions at stage k, the variable xk does not depend on the unknown ξk; instead, it relies on ξk1. According to the wealth dynamic equation, xk is also influenced by the return rate rk1. Therefore, we denote xk(rk1,ξk1) as a variable dependent on both rk1 and ξk1.

    In this section, we derive the analytical optimal solution for the multi-period robust portfolio selection problem using dynamic programming principles.

    First, at stage K, given the cumulative return rK1 from time K1, we define the objective function as the robust CVaR for stage K. The corresponding robust optimization model is formulated as follows:

    VK(rK1)=minxKRnmaxξKDK(μK,Γ)minαKR{αK+11βKE[(RαKxTK(ξKRe))+]} s.t. rK1+(E[ξK])TxK+R(1xTKe)=r_K.

    Since the set Dk is convex and closed, the function Fβk(xk,αk) is also convex. By applying the minimax theorem (see Theorem 4.2 in [35]), we can interchange the order of the maximum and minimum operations. Therefore, we formulate the robust CVaR portfolio selection problem as follows:

    RCVaRβk(xk)=minαkRmaxπkDkFβk(xk,αk)=minαKRαK+11βKmaxξKDK(μK,Γ)[E(RαKxTK(ξKRe))+].

    Using Lemmas 2.2 and 2.4 from [36], we obtain

    RCVaRβk(xk)=minαKRαK+R(xKTe1)αKxKTμK+xKTΓxK+(R(xKTe1)αKxKTμK)22(1βK). (3.1)

    The first-order optimality condition for problem (3.1) is given by

    112(1βK)12(1βK)2(R(xKTe1)αKxKTμK)2xKTΓxK+(R(xKTe1)αKxKTμK)2=0.

    Consequently, we have

    αK=2βK12βK(1βK)xKTΓxKxKTμKR(1xKTe).

    The robust CVaR portfolio selection problem is formulated as follows:

    RCVaRβK(x)=βK1βKxKTΓxKxKTμKR(1xKTe).

    Let ˜μK=μKRe. Consequently, we formulate the following optimization problem:

    VK(rK1)=minxKRnβK1βKxTKΓxKxTK˜μKR s.t. rK1+(E[ξK])TxK+R(1xTKe)=r_K. (3.2)

    Let ξK=E[ξK]Re, and set sK=xTK˜μK. Employing a transformation from the proof of Theorem 2.5 in [36], problem (3.2) is equivalent to

    minsKRminxKRnβK1βKxTKΓxKxTK˜μKR s.t. ξKTxK=r_KrK1R,xTK˜μK=sK. (3.3)

    To proceed, we first solve problem (3.4):

    minxKxTKΓxK s.t. ξKTxK=r_KrK1R,xTK˜μK=sK. (3.4)

    By obtaining the optimal solution xK(sK) for (3.4), we can derive xK(sK)TΓxK(sK) and substitute this into the objective function of problem (3.3), transforming it into an unconstrained optimization problem. This leads to the optimal strategy for period K. The Lagrangian function for problem (3.4) is therefore

    L(xK,λK1,λK2)=xTKΓxK+λK1(sKxTK˜μK)+λK2(r_KrK1RxTKξK),

    and applying the first-order optimality conditions yields the following equations:

    {LxK=2ΓxKλK1˜μKλK2ξK=0,(3.5)xTKξK(r_KrK1R)=0,(3.6)xTK˜μKsK=0.(3.7)

    From (3.5), we obtain

    xK=12Γ1(λK1˜μK+λK2ξK), (3.8)

    and substituting (3.8) into (3.6) and (3.7) gives us

    {(λK1˜μK+λK2ξK)TΓ1˜μK=2sK,(λK1˜μK+λK2ξK)TΓ1ξK=2(r_KrK1R).
    {λK1˜μTKΓ1˜μK+λK2(ξK)TΓ1˜μK=2sK,λK1˜μTKΓ1ξK+λK2(ξK)TΓ1ξK=2(r_KrK1R).

    We define the notation

    aK0:=(ξK)TΓ1ξK,aK1:=(ξK)TΓ1˜μK,aK2:=˜μTKΓ1˜μK,
    dK0:=aK0aK0aK2(aK1)2,dK1:=aK1aK0aK2(aK1)2,dK2:=aK2aK0aK2(aK1)2.

    Then,

    {aK2λK1+aK1λK2=2sK,aK1λK1+aK0λK2=2(r_KrK1R).

    That is,

    (aK2aK1aK1aK0)(λK1λK2)=2(sKr_KrK1R).

    We find the Lagrange multipliers

    (λK1λK2)=2aK0aK2(aK1)2(aK0aK1aK1aK2)(sKr_KrK1R)=2(dK0dK1dK1dK2)(sKr_KrK1R).

    Substituting λK1, λK2 into (3.8), we derive the optimal solution for problem (3.4):

    xK(sK)=12Γ1(˜μKξK)(λK1λK2)=(Γ1˜μKΓ1ξK)(dK0dK1dK1dK2)(sKr_KrK1R). (3.9)

    Thus,

    xK(sK)TΓxK(sK)=((dK0sKdK1(r_KrK1R)dK2(r_KrK1R)dK1sK)(˜μTK(ξK)T))Γ1((˜μKξK)(dK0sKdK1(r_KrK1R)dK2(r_KrK1R)dK1sK))=aK2(dK0sKdK1(r_KrK1R))2+aK0(dK2(r_KrK1R)dK1sK)2+2aK1(dK0dK1s2K+(dK0dK2+(dK1)2)(r_KrK1R)sKdK1dK2(r_KrK1R)2)=(aK2(dK0)22aK1dK0dK1+aK0(dK1)2)(sK)2+2(aK1(dK1)2aK2dK0dK1+aK1dK0dK2aK0dK1dK2)(r_KrK1R)sK+(aK2(dK1)22aK1dK1dK2+aK0(dK2)2)(r_KrK1R)2=dK0(sK)22dK1(r_KrK1R)sK+dK2(r_KrK1R)2. (3.10)

    Substituting (3.10) into the objective function of problem (3.3) transforms it into the following unconstrained optimization problem:

    minsKRhβK(sK):=βK1βKdK0(sK)22dK1(r_KrK1R)sK+dK2(r_KrK1R)2sKR. (3.11)

    The first-order optimality condition for problem (3.11) is

    hβK(sK)=βK1βKdK0sKdK1(r_KrK1R)dK0(sK)22dK1(r_KrK1R)sK+dK2(r_KrK1R)21=0,

    leading to

    βK(dK0sKdK1(r_KrK1R))2=(1βK)(dK0(sK)22dK1(r_KrK1R)sK+dK2(r_KrK1R)2)dK0(βKdK0(1βK))(sK)22dK1(βKdK0(1βK))(r_KrK1R)sK+(βK(dK1)2dK2(1βK))(r_KrK1R)2=0.

    This presents two scenarios:

    (1) If βK1βKdK0>1, the optimal solution for problem (3.11) is

    sK=dK1(r_KrK1R)dK0+(r_KrK1R)dK0(βKdK0(1βK))[βKdK0(1βK)][(dK1)2(βKdK0(1βK))dK0(βK(dK1)2dK2(1βK))]=dK1(r_KrK1R)dK0+(r_KrK1R)dK0(βKdK0(1βK))[βKdK0(1βK)][(dK0dK2(dK1)2)(1βK)]=dK1(r_KrK1R)dK0+(r_KrK1R)dK0βKdK0(1βK)[(dK0dK2(dK1)2)(1βK)]=(r_KrK1R)(dK1dK0+dK0dK2(dK1)2dK0βKdK01βK1).

    (2) If βK1βKdK01, the optimal solution for problem (3.11) is sK=+, indicating that problem (3.11) is unbounded.

    In scenario (1), substituting sK back into (3.9) provides the optimal investment strategy for period K, and substituting into (3.11) yields the optimal objective value for problem (3.2). Specifically, the optimal investment strategy for period K is

    xK=(r_KrK1R)(Γ1˜μKΓ1ξK)(dK0dK1dK1dK2)(dK0dK2(dK0)2dK0βKdK01βK1+dK1dK01).

    Substituting ξK=E[ξK]Re=˜μK into the above expression yields

    xK=(r_KrK1R)Γ1˜μK(11)(dK0dK1dK1dK2)(dK0dK2(dK1)2dK0βKdK01βK1+dK1dK01).

    Remark 1. A higher βk implies greater risk aversion. The ratio βk1βk scales the investor's risk aversion. dk0 inversely measures diversification potential; a smaller dk0 indicates higher diversification. When βk1βkdk01, the level of risk aversion βk is insufficient relative to the market's diversification potential dk0. This imbalance results in unbounded leverage in risky assets to minimize risk or maximize returns, causing the optimization problem to not have a finite solution (i.e., sK=+). When βk1βkdk0>1, risk aversion dominates market conditions, ensuring the existence of a finite optimal portfolio.

    The expression for the optimal solution xK involves ˜μK, which can be represented as ˜μK=μKRe. The random sequence {μk,k=1,,K} forms a Markov chain with m possible states μ1,μ2,,μm. Thus, the derived xK varies with the state of μk, making this optimal solution a strategy that adapts to prior information. Suppose an investor finds that the sub-strategy from time k to K does not achieve optimality based on their initial decision at time k. In that case, it indicates that the initial investment choice is not the most effective across the entire investment period.

    Compared to static investment strategies that fail to adapt to market changes, the adoption of a dynamic decision-making approach aligns more closely with real-world scenarios and represents a superior strategy. However, in practice, market information evolves, and the information available is continuously updated. If investors can swiftly adjust their strategies in response to market fluctuations, they tend to achieve greater returns than relying on a fixed approach. From the above conclusions, the optimal value for period K is given by

    VK(rK1)=βK1βKdK0(sK)22dK1(r_KrK1R)sK+dK2(r_KrK1R)2sKR=βK1βKdK0(sKdK1dK0(r_KrK1R))2+dK0dK2(dK1)2dK0(r_KrK1R)2sKR=dK0βK1βK(dK0dK2(dK1)2(r_KrK1R)dK0βKdK01βK1)2+dK0dK2(dK1)2(dK0)2(r_KrK1R)2sKR=dK0βK1βK(dK0dK2(dK1)2)(r_KrK1R)2(dK0)2(βKdK0βKdK0(1βK))sKR=(r_KrK1R)βK1βKdK0dK2(dK1)2βKdK01βK1dK1(r_KrK1R)dK0dK0dK2(dK1)2(r_KrK1R)dK0βKdK01βK1R=(r_KrK1R)(dK0dK2(dK1)2βKdK01βK1dK0dK1dK0)R=qK1(r_KrK1R)R, (3.12)

    where

    qK1=(dK0dK2(dK1)2βKdK01βK1dK0dK1dK0).

    This notation is convenient and aids in deriving recursive relationships during the solving process.

    When the investor is in period K2, for a given cumulative return rate rK2, substituting (3.12) into the objective function of problem (2.1) allows us to express the corresponding optimal investment decision problem as

    VK1(rK2)=minxK1{βK11βK1xTK1ΓxK1(μK1Re)TxK1R+E[VK(wK1)]} s.t. rK2+(E[ξK1])TxK1+R(1xTK1e)=r_K1. (3.13)

    Substituting the expression VK(rK1) into the objective function of problem (3.13) yields

    VK1(rK2)=minxK1{βK11βK1xTK1ΓxK1(μK1Re)TxK1R+qK1E[(r_KrK1R)]R} s.t. rK2+(E[ξK1])TxK1+R(1xTK1e)=r_K1.

    Incorporating the constraints into the above objective function, problem (3.13) becomes

    VK1(rK2)=minxK1{βK11βK1xTK1ΓxK1(μK1Re)TxK1R+qK1(r_K(rK2+(E[ξK1])TxK1+R(1xTK1e))R)R}=minxK1{βK11βK1xTK1ΓxK1(˜μK1+qK1ξK1)TxK1+qK1(r_KrK2)2R(qK1+1)}.

    Letting sK1=xTK1(˜μK1+qK1ξK1) and employing a method similar to that used for the period K problem, we can equivalently represent problem (3.13) as

    VK1(rK2)=minsK1RminxK1Rn{βK11βK1xTK1ΓxK1sK1+qK1(r_KrK2)2R(qK1+1)} s.t. rK2+(E[ξK1])TxK1+R(1xTK1e)=r_K1,xTK1(˜μK1+qK1ξK1)=sK1. (3.14)

    To this end, we first solve problem (3.15)

    minxK1RnxTK1ΓxK1 s.t. xTK1ξK1=r_K1rK2R,xTK1(˜μK1+qK1ξK1)=sK1. (3.15)

    By finding the optimal solution xK1(sK1) for (3.15), we can derive xK1(sK1)TΓ xK1(sK1) and substitute it into the objective function of problem (3.14), thus transforming it into an unconstrained optimization problem to determine the optimal strategy for period K1. The Lagrangian function for problem (3.15) is

    L(xK1,λK11,λK12)=xTK1ΓxK1+λK11(sK1xTK1(˜μK1+qK1ξK1))+λK12(r_K1rK2RxTK1ξK1).

    Applying the first-order optimality conditions yields the following equations:

    {LxK1=2ΓxK1λK11(˜μK1+qK1ξK1)λK12ξK1=0,(3.16)xTK1ξK1(r_K1rK2R)=0,(3.17)xTK1(˜μK1+qK1ξK1)sK1=0.(3.18)

    From (3.16), we obtain

    xK1=12Γ1(λK11(˜μK1+qK1ξK1)+λK12ξK1), (3.19)

    and substituting (3.19) into (3.17) and (3.18) yields

    {(λK11(˜μK1+qK1ξK1)+λK12ξK1)TΓ1(˜μK1+qK1ξK1)=2sK1,(λK11(˜μK1+qK1ξK1)+λK12ξK1)TΓ1ξK1=2(r_K1rK2R).
    {(aK12+2qK1aK11+q2K1aK10)λK11+(aK11+qK1aK10)λK12=2sK1,(aK11+qK1aK10)λK11+aK10λK12=2(r_K1rK2R).

    Thus, (aK12+2qK1aK11+q2K1aK10aK11+qK1aK10aK11+qK1aK10aK10)(λK11λK12)=2(sK1r_K1rK2R).

    The Lagrange multipliers are

    (λK11λK12)=2aK10aK12(aK11)2(aK10(aK11+qK1aK10)(aK11+qK1aK10)aK12+2qK1aK11+q2K1aK10)(sK1r_K1rK2R).

    Substituting into (3.19) and simplifying yields the optimal solution for problem (3.15):

    xK1(sK1)=12Γ1(˜μK1+qK1ξK1ξK1)(λK11λK12)=(Γ1(˜μK1+qK1ξK1)Γ1ξK1)(dK10(dK11+qK1dK10)(dK11+qK1dK10)dK12+2qK1dK11+q2K1dK10)(sK1r_K1rK2R). (3.20)

    Consequently, (aK10aK12(aK11)2)2xK1(sK1)TΓxK1(sK1)=

    ((aK10sK1(aK11+qK1aK10)(r_K1rK2R)(aK12+2qK1aK11+q2K1aK10)(r_K1rK2R)(aK11+qK1aK10)sK1)((˜μK1+qK1ξK1)T(ξK1)T))Γ1((˜μK1+qK1˜ξK1ξK1)(aK10sK1(aK11+qK1aK10)(r_K1rK2R)(aK12+2qK1aK11+q2K1aK10)(r_K1rK2R)(aK11+qK1aK10)sK1))=[(aK12+2qK1aK11+q2K1aK10)(aK10)2aK10(aK11+qK1aK10)2]s2K1+2[(aK11+qK1aK10)3aK10(aK11+qK1aK10)(aK12+2qK1aK11+q2K1aK10)](r_K1rK2R)sK1+[aK10(aK12+2qK1aK11+q2K1aK10)2(aK12+2qK1aK11+q2K1aK10)(aK11+qK1aK10)2](r_K1rK2R)2=[aK10aK12(aK11)2][aK10s2K12(aK11+qK1aK10)(r_K1rK2R)sK1+(aK12+2qK1aK11+q2K1aK10)(r_K1rK2R)2].

    Hence,

    xK1(sK1)TΓxK1(sK1)=dK10s2K12(dK11+qK1dK10)(r_K1rK2R)sK1+(dK12+2qK1dK11+q2K1dK10)(r_K1rK2R)2. (3.21)

    Therefore, substituting (3.21) into the objective function of problem (3.14) reveals that problem (3.14) is equivalent to the unconstrained optimization problem

    minsK1RhβK1(sK1):=βK11βK1dK10s2K12(dK11+qK1dK10)(r_K1rK2R)sK1¯+(dK12+2qK1dK11+q2K1dK10)(r_K1rK2R)2sK1+qK1(r_KrK2)2R(qK1+1). (3.22)

    The first-order optimality condition for problem (3.22) is

    hβK1(sK1)=βK11βK1dK10sK1(dK11+qK1dK10)(r_K1rK2R)dK10s2K12(dK11+qK1dK10)(r_K1rK2R)sK1+(dK12+2qK1dK11+q2K1dK10)(r_K1rK2R)21=0. (3.23)

    Thus, we have

    βK1(dK10sK1(dK11+qK1dK10)(r_K1rK2R))2=(1βK1)(dK10s2K12(dK11+qK1dK10)(r_K1rK2R)sK1+(dK12+2qK1dK11+q2K1dK10)(r_K1rK2R)2)dK10(βK1dK10(1βK1))s2K12(dK11+qK1dK10)(βK1dK10(1βK1))(r_K1rK2R)sK1+(βK1(dK11+qK1dK10)2(dK12+2qK1dK11+q2K1dK10)(1βK1))(r_K1rK2R)2=0.

    We then consider two scenarios:

    (1) When βK11βK1dK10>1, the optimal solution for problem (3.22) is

    sK1=(dK11+qK1dK10)(r_K1rK2R)dK10+(r_K1rK2R)dK10βK1dK10(1βK1)(dK10(dK12+2qK1dK11+q2K1dK10)(dK11+qK1dK10)2)(1βK1)=(dK11+qK1dK10)(r_K1rK2R)dK10+dK10(dK12+2qK1dK11+q2K1dK10)(dK11+qK1dK10)2dK10βK1dK10(1βK1)1(r_K1rK2R).

    (2) When βK11βK1dK101, the optimal solution is sK1=+, indicating that problem (3.22) is unbounded.

    In scenario (1), substituting sK1 into (3.19) yields the optimal investment strategy for period K1, and substituting into (3.14) provides the optimal objective value for problem (3.13). Thus, the optimal value for period K1 is

    VK1(rK2)=(r_K1rK2R)dK10(dK12+2qK1dK11+q2K1dK10)(dK11+qK1dK10)2βK1dK101βK11dK10(dK11+qK1dK10)(r_K1rK2R)dK10+qK1(r_KrK2)2R(qK1+1)=(r_K1rK2R)(dK10(dK12+2qK1dK11+q2K1dK10)(dK11+qK1dK10)2βK1dK101βK11dK11qK1dK10dK10)+qK1(r_KrK2)2R(qK1+1)=qK2(r_K1rK2)+qK1(r_KrK2)2R(12qK2+qK1+1),

    where qK2=

     dK10(dK12+2qK1dK11+q2K1dK10)(dK11+qK1dK10)2βK1dK101βK11dK11qK1dK10dK10, qK1=dK0dK2(dK1)2βKdK01βK1dK0dK1dK0.

    The optimal investment strategy for period K1 is

    xK1=(r_K1rK2R)(Γ1(˜μK1+qK1ξK1)Γ1ξK1)(dK10(dK11+qK1dK10)(dK11+qK1dK10)dK12+2qK1dK11+q2K1dK10)((dK11+qK1dK10)dK10+dK10(dK12+2qK1dK11+q2K1dK10)(dK11+qK1dK10)2dK10βK1dK10(1βK1)11).

    By substituting ξK=E[ξK]Re=˜μK into the expression, we find

    xK1=(r_K1rK2R)Γ1˜μK1(1+qK11)(dK10(dK11+qK1dK10)(dK11+qK1dK10)dK12+2qK1dK11+q2K1dK10)((dK11+qK1dK10)dK10+dK10(dK12+2qK1dK11+q2K1dK10)(dK11+qK1dK10)2dK10βK1dK10(1βK1)11).

    We apply the same method from period K down to period 1 to solve the optimal investment decision problems for each subsequent stage. Theorem 1 summarizes the resulting optimal strategies.

    Theorem 1. For all k=1,,K, let ξk=E[ξk]Re, ak0:=(ξk)TΓ1ξk, ak1:=(ξk)TΓ1˜μk, ak2:=˜μTkΓ1˜μk, dk0:=ak0ak0ak2(ak1)2, dk1:=ak1ak0ak2(ak1)2, dk2:=ak2ak0ak2(ak1)2,

    qi=di+10(di+12+2di+11K1j=i+1qj+di+10(K1j=i+1qj)2)(di+11+di+10K1j=i+1qj)2βi+1di+101βi+11di+10di+11+(K1j=i+1qj)di+10di+10.

    If βk1βkdk0>1 for all k=1,,K, then, given a cumulative return rk1, the optimal objective value and optimal investment strategy at stage k are

    Vk(rk1)=Ki=kqi1(r_irk1)R(Kj=k(jk+1)qj1+Kk+1)

    and

    xk=(r_krk1R)Γ1˜μk(1+K1j=kqj1)(dk0(dk1+dk0K1j=kqj)(dk1+dk0K1j=kqj)dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)((dk1+dk0K1j=kqj)dk0+dk0(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(dk1+dk0K1j=kqj)2dk0βkdk0(1βk)11),

    respectively. If βk1βkdk01 for some k (1kK), the optimal solution at period k diverges to infinity, making problem (2.1) unbounded.

    Proof. Assume the optimal value and solution hold at stage k+1. At stage k, given the cumulative return rk1, the corresponding optimal decision problem is expressed as

    Vk(rk1)=minxkRnβk1βkxTkΓxk(μkRe)TxkR+qk(r_k+1E[rk])++qK2(r_K1E[rk])+qK1(r_KE[rk])R(Kj=k+1(jk)qj1+Kk) s.t. rk1+(E[ξk])Txk+R(1xTke)=r_k. (4.1)

    Substituting the constraints into the objective function of the problem (4.1), we obtain

    Vk(rk1)=minxkRnβk1βkxTkΓxk˜μTkxk+qk(r_k+1(rk1+(E[ξk])Txk+R(1xTke)))++qK2(r_K1(rk1+(E[ξk])Txk+R(1xTke)))+qK1(r_K(rk1+(E[ξk])Txk+R(1xTke)))R(Kj=k+1(jk)qj1+Kk+1)=minxkRnβk1βkxTkΓxk(˜μk+(qk+qk+1++qK1)ξk)Txk+qk(r_k+1rk1)++qK2(r_K1rk1)+qK1(r_Krk1)R(Kj=k+1(jk)qj1+Kk+1),

    with sk=xTk(˜μk+(qk+qk+1++qK1)ξk) showing that problem (4.1) is equivalent to

    minskRminxkRnβk1βkxTkΓxksk+qk(r_k+1rk1)++qK2(r_K1rk1)+qK1(r_Krk1)R(Kj=k+1(jk)qj1+Kk+1) s.t. xTk(˜μk+(qk+qk+1++qK1)ξk)=sk. (4.2)

    To this end, we consider the following problem:

    minxkRnxTkΓxk s.t. xTkξk=r_krk1R,xTk(˜μk+K1j=kqjξk)=sk. (4.3)

    The Lagrangian function for problem (4.3) is

    L(xk,λk1,λk2)=xTkΓxk+λk1(skxTk(˜μk+K1j=kqjξk))+λk2(r_krk1RxTk˜ξk).

    Using the first-order optimality conditions, we derive the following equations:

    {Lxk=2Γxkλk1(˜μk+K1j=kqjξk)λk2ξk=0,(4.4)xTkξk(r_krk1R)=0,(4.5)xTk(˜μk+K1j=kqjξk)sk=0.(4.6)

    From (4.4), we have

    xk=12Γ1(λk1(˜μk+K1j=kqjξk)+λk2ξk), (4.7)

    substituting (4.7) into (4.5) and (4.6) yields

    (ak2+2ak1K1j=kqj+ak0(K1j=kqj)2ak1+ak0K1j=kqjak1+ak0K1j=kqjak0)(λk1λk2)=2(skr_krk1R).

    Thus, the Lagrange multipliers are

    (λk1λk2)=(dk0(dk1+dk0K1j=kqj)(dk1+dk0K1j=kqj)dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(skr_krk1R).

    Substituting λk1, λk2 into (4.7) and simplifying gives the optimal solution for problem (4.3):

    xk(sk)=(Γ1(˜μk+K1j=kqjξk)Γ1ξk)(dk0(dk1+dk0K1j=kqj)(dk1+dk0K1j=kqj)dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(skr_krk1R). (4.8)

    Therefore,

    (xk(sk))TΓxk(sk)=dk0s2k2(dk1+dk0K1j=kqj)(r_krk1R)sk+(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(r_krk1R)2. (4.9)

    Substituting (4.9) into the objective function of problem (4.2) transforms the problem into an unconstrained optimization problem

    minskRhβk(sk):=βk1βkdk0s2k2(dk1+dk0K1j=kqj)(r_krk1R)sk+(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(r_krk1R)2sk+qk(r_k+1rk1)++qK2(r_K1rk1)+qK1(r_Krk1)R(Kj=k+1(jk)qj1+Kk+1). (4.10)

    At this point, we consider two cases: (1) When βk1βkdk0>1,

    sk=(dk1+dk0K1j=kqj)(r_krk1R)dk0+(r_krk1R)dk0βkdk0(1βk)(dk0(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(dk1+dk0K1j=kqj)2)(1βk),

    that is

    sk=(dk1+dk0K1j=kqj)(r_krk1R)dk0+dk0(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(dk1+dk0K1j=kqj)2dk0βkdk0(1βk)1(r_krk1R).

    (2) When βk1βkdk01, the optimal solution for problem (4.10) is sk=+, indicating it is unbounded. In case (1), substituting sk back into (4.8) provides the optimal investment strategy for period k, and substituting into (4.2) gives the optimal objective value for problem (4.1). Thus, the optimal value for period k is

    Vk=(r_krk1R)(dk0(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(dk1+dk0K1j=kqj)2βkdk01βk1dk0dk1+dk0K1j=kqjdk0)+qk(r_k+1rk1)++qK2(r_K1rk1)+qK1(r_Krk1)R(Kj=k+1(jk)qj1+Kk+1)=qk1(r_krk1R)+qk(r_k+1rk1)++qK2(r_K1rk1)+qK1(r_Krk1)R(Kj=k+1(jk)qj1+Kk+1),

    where qk1=

    (dk0(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(dk1+dk0K1j=kqj)2βkdk01βk1dk0dk1+Rdk0K1j=kqjRjkdk0).

    The optimal investment strategy for period k is

    xk=(r_krk1R)Γ1˜μk(1+K1j=kqj1)(dk0(dk1+dk0K1j=kqj)(dk1+dk0K1j=kqj)dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)((dk1+dk0K1j=kqj)dk0+dk0(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(dk1+dk0K1j=kqj)2dk0βkdk0(1βk)11).

    Thus, by employing dynamic programming, we can recursively solve problem (2.1) to obtain the optimal objective values and strategies for period k (where k=1,,K) as

    Vk(wk1)=Ki=kqi1(r_irk1)R(Kj=k(jk+1)qj1+Kk+1).

    Here,

    qi=di+10(di+12+2di+11K1j=i+1qj+di+10(K1j=i+1qj)2)(di+11+di+10K1j=i+1qj)2βi+1di+101βi+11di+10di+11+(K1j=i+1qj)di+10di+10,

    and

    xk=(r_krk1R)Γ1˜μk(1+K1j=kqj1)(dk0(dk1+dk0K1j=kqj)(dk1+dk0K1j=kqj)dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)((dk1+dk0K1j=kqj)dk0+dk0(dk2+2dk1K1j=kqj+dk0(K1j=kqj)2)(dk1+dk0K1j=kqj)2dk0βkdk0(1βk)11).

    With the analytical solution provided in Theorem 1, the optimal investment strategies can be directly determined for portfolio selection at each stage within a robust optimization framework.

    Our dataset consists of five stocks (NVDA, DFS, MTCH, WBA, and GOOGL) from https://finance.yahoo.com. These stocks were selected for sectoral diversity across technology, finance, healthcare, and consumer discretionary sector while also exhibiting high liquidity and non-Gaussian return characteristics, as validated by the Shapiro-Wilk test in Table 1. It includes the weekly closing price of these stocks from January 5, 2014, to January 7, 2024, covering diverse market regimes, such as bull markets and the COVID-19 crash. This selection ensures rigorous validation of the regime-switching CVaR model and aligns with our focus on distributionally robust optimization under uncertain return distributions. Figure 1 presents the time series plots of the original price data. This study adopts the robust CVaR as the risk measure to solve the robust optimization problem, aiming to minimize risk while satisfying the minimum target return rate constraint.

    Table 1.  Moments and KS test statistic.
    mean Var skew kurt S-W test P-value
    NVDA 0.0113 0.0036 0.4364 2.005 0.9771 0.0000
    DFS 0.0031 0.0034 0.5131 27.1993 0.7765 0.0000
    MTCH 0.0032 0.0040 0.3137 3.7597 0.9564 0.0000
    WBA -0.0010 0.0016 -0.0633 1.3410 0.9842 0.0000
    GOOGL 0.0038 0.0014 0.6776 4.5066 0.9609 0.0000

     | Show Table
    DownLoad: CSV
    Figure 1.  Time series plots of the dataset.

    We computed the logarithmic return rates for all five stocks. Table 1 presents the central moments and Shapiro-Wilk test results for the selected assets. Based on the S-W test results, the P-values in the last column of Table 1 were all below 0.01. Thus, the null hypothesis of normality is rejected, indicating that the data do not follow a normal distribution.

    We aim to dynamically adjust asset allocation based on capital market conditions. The underlying regimes are predicted by a Markov chain with two regimes, which are commonly interpreted as a bear market or a bull market, denoted as U={0,1}. By incorporating regime switching, the model can more accurately reflect market dynamics.

    Using historical data, we identify market regimes based on a threshold criterion. Specifically, we set a threshold of 0.01, meaning a return above 1% signifies a bull market. Asset returns, volatility, and other parameters fluctuate between the two regimes. We estimate the expected mean returns and covariance matrices for each regime based on historical data, as shown in Eq (5.1) to (5.4). The confidence level βk is set to 0.95, which is assumed to be the same across regimes. The minimum required return r_k is set to 0.03, and the risk-free rate R is set to 0.02.

    μ0=102(0.78430.51381.11340.80890.2239)T, (5.1)
    μ1=102(0.99480.31960.48030.00120.3353)T, (5.2)
    Γ0=102(0.61540.34710.39450.03840.23180.34711.32600.45500.22380.23170.39450.45500.97930.11170.20200.03840.22380.11170.38970.06340.23180.23170.20200.06340.2938), (5.3)
    Γ1=103(2.76410.53270.87170.48350.73270.53270.85970.28030.41150.38510.87170.28032.25740.33980.57780.48350.41150.33980.98860.33190.73270.38510.57780.33190.9563). (5.4)

    By analyzing regime transition frequencies, we compute the transition probability matrix

    P=(0.58230.41770.63900.3610).

    The optimal investment strategy can be determined using dynamic programming, which recursively simplifies the problem to reduce computational complexity, particularly when state transitions exhibit Markov properties. The state variables include the current market condition and current wealth levels. The decision variable is the portfolio adjustments at each stage. State transitions are determined by the transition probability matrix of the market states, while the objective function represents the minimum risk from the current stage to the final stage.

    A robust optimization model that minimizes risk under the worst-case scenario is applied at each stage. Based on the current state and decision-making, the optimal portfolios for the next stage are computed while considering the state transition probabilities. Thus, embedding the robust optimization problem within the dynamic programming framework is necessary.

    By recursively decomposing the multi-stage problem through dynamic programming, the robust optimization subproblem is solved at each stage by calculating backward from the final stage k=K to the initial time k=1. The proposed framework's time complexity is O(Km2n3), where K is the number of periods, m is the number of regimes, and n is the number of risky assets. Dynamic programming avoids the exponential explosion of the scenario tree, enhancing the computational efficiency of the multi-stage robust portfolio optimization problem.

    We develop an investment strategy that dynamically adjusts asset allocation in response to regime transitions. Table 2 presents the optimal portfolios across different stages, demonstrating a dynamic adjustment strategy that minimizes risk while satisfying return constraints. The initial market state is assumed to be a bull market with an initial wealth of 10.0. At each stage, μk is dynamically updated, and the target value Vk and strategy are adjusted accordingly. The total number of stages is set to K=10. The sum of portfolio weights in each row is less than 1.0 due to the inclusion of the risk-free asset. The changes in portfolio allocations across different stages are examined, with allocations to certain assets increasing while others decrease, reflecting the model's expectations of regime switches.

    Table 2.  Optimal portfolios under DP.
    Period Regime NVDA DFS MTCH WBA GOOGL Risk Asset Weight
    0 1 0.2000 0.2000 0.2000 0.2000 1.6912×109 0.8000
    1 0 0.2000 0.2000 1.1299×1010 1.2979×1010 2.3682×1010 0.4000
    2 0 3.2500×1011 0.2000 6.0090×1011 0.2000 0.2000 0.6000
    3 0 1.2931×107 4.6059×1010 0.2000 9.7313×1010 0.2000 0.4000
    4 1 0.2000 1.0664×109 9.9738×1010 0.2000 0.2000 0.6000
    5 1 0.2000 0.2000 0.2000 1.7060×1011 3.3000×1011 0.6000
    6 0 0.2000 4.8124×1010 3.5599×1010 1.1482×109 3.9763×1010 0.2000
    7 1 2.5804×1010 0.2000 0.2000 0.2000 9.1840×1010 0.6000
    8 0 0.2000 0.2000 1.8773×109 0.2000 0.2000 0.8000
    9 0 0.2000 1.1065×1010 9.4690×1010 3.4756×1010 0.2000 0.4000

     | Show Table
    DownLoad: CSV

    Building upon the optimal portfolios derived from the dynamic programming framework as detailed in Table 2, we further evaluate the strategy's efficacy and robustness. Key performance indicators, including the Sharpe ratio (SR), compound annual growth rate (CAGR), maximum drawdown (MDD), and turnover ratio (TR), are reported in Table 3.

    Table 3.  Performance measures of the portfolios.
    Periods SR CAGR MDD TR
    10 1.1235 3.2659% 12.7198% 37.1429%

     | Show Table
    DownLoad: CSV

    The strategy exhibits robust performance over a 10-period investment horizon, achieving a Sharpe ratio of 1.12 and an annualized CAGR of 3.27%, indicating consistently superior risk-adjusted returns and stable growth. The maximum drawdown of 12.72% demonstrates effective downside risk management, while the turnover rate of 37.14% indicates that a balanced portfolio rebalancing corresponds with transaction costs. These performance measures provide empirical validation that our model effectively addresses the multi-stage robust portfolio selection problem under regime switching.

    In this study, we propose a dynamic multi-period robust portfolio selection framework that integrates regime-switching techniques and distributionally robust optimization under CVaR risk measures. By refining the uncertainty set with known first and second moments, we construct a dynamic model that captures the dependencies between consecutive periods. Our approach employs dynamic programming to address the robust optimization problem, ensuring that the resulting optimal investment strategies adapt dynamically based on the state of the uncertainty set. Compared to static strategies that fail to respond to market fluctuations, our approach better aligns with real-world conditions and offers a more effective solution. Leveraging convex duality and dynamic programming, we derive analytical optimal investment strategies that dynamically adjust allocations based on regimes. This framework develops multi-period portfolio optimization by integrating dynamic uncertainty modeling with robust risk management.

    Although we have analytically solved the multi-stage robust portfolio selection problem, the complexity of the problem increases with the number of stages. We assume the Markov transition matrices are time-homogeneous, while structural breaks on asset returns vary across markets. Developing a robust adaptive transition estimator will be a potential research direction. There remain numerous questions for further study. This study considers the case where the distribution of asset returns is uncertain. Future research could extend this framework to scenarios where both the distribution and mean of the return are uncertain. Approaches employing CVaR and VaR as risk measures would be valuable, especially in scenarios where the mean belongs to an ellipsoidal uncertainty set or a Wasserstein ambiguity set. Identifying multi-period risk measure models that not only have a strong financial and economic foundation but also ensure practical applicability remains a critical research area. Future work will apply Dempster-Shafer [37] and multi-scale fusion [38] to optimize dynamic CVaR in regime-switching portfolios. Furthermore, enhancing computational efficiency in solving multi-period portfolio selection problems through stochastic programming and other relevant methodologies, such as PMCTNN [39], will be crucial for advancing multi-period investment strategies.

    The author declares she has not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by the Science and Technology Research Program of Chongqing Municipal Education Commission of China (Grant No. KJQN202201613).

    The author declares no conflict of interest.

    [1] Kultz D (2005) Molecular and evolutionary basis of the cellular stress response. Annu Rev Physiol 67: 225-257. doi: 10.1146/annurev.physiol.67.040403.103635
    [2] Nassif M, Matus S, Castillo K, et al. (2010) Amyotrophic lateral sclerosis pathogenesis: a journey through the secretory pathway. Antioxid Redox Sign 13: 1955-1989. doi: 10.1089/ars.2009.2991
    [3] Schapira AH, Olanow CW, Greenamyre JT, et al. (2014) Slowing of neurodegeneration in Parkinson's disease and Huntington's disease: future therapeutic perspectives. Lancet 384: 545-555.
    [4] Massano J, Bhatia KP (2012) Clinical approach to Parkinson's disease: features, diagnosis, and principles of management. Cold Spring Harbor Perspect Med 2: a008870.
    [5] Chaudhuri KR, Odin P, Antonini A, et al. (2011) Parkinson's disease: the non-motor issues. Parkinsonism Relat D 17: 717-723. doi: 10.1016/j.parkreldis.2011.02.018
    [6] Greenamyre JT, Hastings TG (2004) Biomedicine. Parkinson's--divergent causes, convergent mechanisms. Science 304: 1120-1122.
    [7] Spillantini MG, Schmidt ML, Lee VM, et al. (1997) Alpha-synuclein in Lewy bodies. Nature 388: 839-840.
    [8] Baba M, Nakajo S, Tu PH, et al. (1998) Aggregation of alpha-synuclein in Lewy bodies of sporadic Parkinson's disease and dementia with Lewy bodies. Am J Pathol 152: 879-884.
    [9] Cox D, Carver JA, Ecroyd H (2014) Preventing alpha-synuclein aggregation: the role of the small heat-shock molecular chaperone proteins. BBA 1842: 1830-1843.
    [10] Bonifati V, Rizzu P, van Baren MJ, et al. (2003) Mutations in the DJ-1 gene associated with autosomal recessive early-onset parkinsonism. Science 299: 256-259.
    [11] Andersson FI, Werrell EF, McMorran L, et al. (2011) The effect of Parkinson's-disease-associated mutations on the deubiquitinating enzyme UCH-L1. J Mol Biol 407: 261-272. doi: 10.1016/j.jmb.2010.12.029
    [12] Dauer W, Przedborski S (2003) Parkinson's disease: mechanisms and models. Neuron 39: 889-909. doi: 10.1016/S0896-6273(03)00568-3
    [13] Dawson TM, Dawson VL (2010) The role of parkin in familial and sporadic Parkinson's disease. Movement Disord 25: S32-39. doi: 10.1002/mds.22798
    [14] Sidransky E, Lopez G (2012) The link between the GBA gene and parkinsonism. Lancet Neurol 11: 986-998.
    [15] Al-Chalabi A, Jones A, Troakes C, et al. (2012) The genetics and neuropathology of amyotrophic lateral sclerosis. Acta neuropathol 124: 339-352.
    [16] Rosen DR, Siddique T, Patterson D, et al. (1993) Mutations in Cu/Zn superoxide dismutase gene are associated with familial amyotrophic lateral sclerosis. Nature 362: 59-62. doi: 10.1038/362059a0
    [17] Neumann M, Sampathu DM, Kwong LK, et al. (2006) Ubiquitinated TDP-43 in frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Science 314: 130-133.
    [18] Arai T, Hasegawa M, Akiyama H, et al. (2006) TDP-43 is a component of ubiquitin-positive tau-negative inclusions in frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Biochem Biophys Res Commun 351: 602-611. doi: 10.1016/j.bbrc.2006.10.093
    [19] Deng HX, Zhai H, Bigio EH, et al. (2010) FUS-immunoreactive inclusions are a common feature in sporadic and non-SOD1 familial amyotrophic lateral sclerosis. Annals Neurol 67: 739-748.
    [20] Nishimura AL, Mitne-Neto M, Silva HC, et al. (2004) A mutation in the vesicle-trafficking protein VAPB causes late-onset spinal muscular atrophy and amyotrophic lateral sclerosis. Am J Hum Genet 75: 822-831.
    [21] Parkinson N, Ince PG, Smith MO, et al. (2006) ALS phenotypes with mutations in CHMP2B (charged multivesicular body protein 2B). Neurology 67: 1074-1077. doi: 10.1212/01.wnl.0000231510.89311.8b
    [22] Deng HX, Chen W, Hong ST, et al. (2011) Mutations in UBQLN2 cause dominant X-linked juvenile and adult-onset ALS and ALS/dementia. Nature 477: 211-215. doi: 10.1038/nature10353
    [23] Johnson JO, Mandrioli J, Benatar M, et al. (2010) Exome sequencing reveals VCP mutations as a cause of familial ALS. Neuron 68: 857-864. doi: 10.1016/j.neuron.2010.11.036
    [24] Maruyama H, Morino H, Ito H, et al. (2010) Mutations of optineurin in amyotrophic lateral sclerosis. Nature 465: 223-226. doi: 10.1038/nature08971
    [25] Fecto F, Yan J, Vemula SP, et al. (2011) SQSTM1 mutations in familial and sporadic amyotrophic lateral sclerosis. Arch Neurol 68:1440-1446. doi: 10.1001/archneurol.2011.250
    [26] Rubino E, Rainero I, Chio A, et al. (2012) SQSTM1 mutations in frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Neurology 79: 1556-1562. doi: 10.1212/WNL.0b013e31826e25df
    [27] Teyssou E, Takeda T, Lebon V, et al. (2013) Mutations in SQSTM1 encoding p62 in amyotrophic lateral sclerosis: genetics and neuropathology. Acta Neuropathol 125: 511-522. doi: 10.1007/s00401-013-1090-0
    [28] Li J, Li W, Jiang ZG, et al. (2013) Oxidative stress and neurodegenerative disorders. Int J Mol Sci 14: 24438-24475. doi: 10.3390/ijms141224438
    [29] Ayala A, Munoz MF, Arguelles S (2014) Lipid peroxidation: production, metabolism, and signaling mechanisms of malondialdehyde and 4-hydroxy-2-nonenal. Oxidative Med Cell Longev: 360438.
    [30] Gandhi S, Abramov AY (2012) Mechanism of oxidative stress in neurodegeneration. Oxidative Med Cell Longev: 428010.
    [31] Halliwell B (2001) Role of free radicals in the neurodegenerative diseases. Drug Aging 18: 685-716 doi: 10.2165/00002512-200118090-00004
    [32] Halliwell B (2006) Oxidative stress and neurodegeneration: where are we now? J Neurochem 97: 1634-1658. doi: 10.1111/j.1471-4159.2006.03907.x
    [33] Milani P, Ambrosi G, Gammoh O, et al. (2013) SOD1 and DJ-1 converge at Nrf2 pathway: a clue for antioxidant therapeutic potential in neurodegeneration. Oxidative Med Cell Longev:836760.
    [34] Parakh S, Spencer DM, Halloran MA, et al. (2013) Redox regulation in amyotrophic lateral sclerosis. Oxidative Med Cell Longev: 408681.
    [35] Streck EL, Czapski GA, Goncalves et al. (2013) Neurodegeneration, mitochondrial dysfunction, and oxidative stress. Oxidative Med Cell Longev: 826046.
    [36] Varcin M, Bentea E, Michotte Y, et al. (2012) Oxidative stress in genetic mouse models of Parkinson's disease. Oxidative Med Cell Longev: 624925.
    [37] Navarro A, Boveris A, Bandez MJ, et al. (2009) Human brain cortex: mitochondrial oxidative damage and adaptive response in Parkinson disease and in dementia with Lewy bodies. Free Radical Biol Med 46: 1574-1580. doi: 10.1016/j.freeradbiomed.2009.03.007
    [38] Alam ZI, Jenner A, Daniel SE, et al. (1997) Oxidative DNA damage in the parkinsonian brain: an apparent selective increase in 8-hydroxyguanine levels in substantia nigra. J Neurochem 69: 1196-1203.
    [39] Abe T, Isobe C, Murata T, et al. (2003) Alteration of 8-hydroxyguanosine concentrations in the cerebrospinal fluid and serum from patients with Parkinson's disease. Neurosci Lett 336: 105-108. doi: 10.1016/S0304-3940(02)01259-4
    [40] Kikuchi A, Takeda A, Onodera H, et al. (2002) Systemic increase of oxidative nucleic acid damage in Parkinson's disease and multiple system atrophy. Neurobiol Dis 9: 244-248. doi: 10.1006/nbdi.2002.0466
    [41] Isobe C, Abe T, Terayama Y (2010) Levels of reduced and oxidized coenzyme Q-10 and 8-hydroxy-2'-deoxyguanosine in the cerebrospinal fluid of patients with living Parkinson's disease demonstrate that mitochondrial oxidative damage and/or oxidative DNA damage contributes to the neurodegenerative process. Neurosci Lett 469: 159-163. doi: 10.1016/j.neulet.2009.11.065
    [42] Nikam S, Nikam P, Ahaley SK, et al. (2009) Oxidative stress in Parkinson's disease. Indian J Clin Biochem 24: 98-101. doi: 10.1007/s12291-009-0017-y
    [43] Barber SC, Mead RJ, Shaw PJ (2006) Oxidative stress in ALS: a mechanism of neurodegeneration and a therapeutic target. Biochim Biophys Acta 1762: 1051-1067. doi: 10.1016/j.bbadis.2006.03.008
    [44] Barber SC, Shaw PJ (2010) Oxidative stress in ALS: key role in motor neuron injury and therapeutic target. Free Radical Boil Med 48: 629-641. doi: 10.1016/j.freeradbiomed.2009.11.018
    [45] Ferrante RJ, Browne SE, Shinobu LA, et al. (1997) Evidence of increased oxidative damage in both sporadic and familial amyotrophic lateral sclerosis. J Neurochem 69: 2064-2074.
    [46] Cutler RG, Pedersen WA, Camandola S (2002) Evidence that accumulation of ceramides and cholesterol esters mediates oxidative stress-induced death of motor neurons in amyotrophic lateral sclerosis. Ann Neurol 52: 448-457. doi: 10.1002/ana.10312
    [47] Pedersen WA, Fu W, Keller JN, et al. (1998) Protein modification by the lipid peroxidation product 4-hydroxynonenal in the spinal cords of amyotrophic lateral sclerosis patients. Ann Neurol 44: 819-824.
    [48] Abe K, Pan LH, Watanabe M, et al. (1995) Induction of nitrotyrosine-like immunoreactivity in the lower motor neuron of amyotrophic lateral sclerosis. Neurosci Lett 199: 152-154. doi: 10.1016/0304-3940(95)12039-7
    [49] Beal MF, Ferrante RJ, Browne SE, et al. (1997) Increased 3-nitrotyrosine in both sporadic and familial amyotrophic lateral sclerosis. Ann Neurol 42: 644-654. doi: 10.1002/ana.410420416
    [50] Shaw PJ, Ince PG, Falkous G, et al. (1995) Oxidative damage to protein in sporadic motor neuron disease spinal cord. Ann Neurol 38: 691-695. doi: 10.1002/ana.410380424
    [51] Fitzmaurice PS, Shaw IC, Kleiner HE, et al. (1996) Evidence for DNA damage in amyotrophic lateral sclerosis. Muscle Nerve 19: 797-798.
    [52] Said Ahmed M, Hung WY, Zu JS, et al. (2000) Increased reactive oxygen species in familial amyotrophic lateral sclerosis with mutations in SOD1. J neurol Sci 176: 88-94. doi: 10.1016/S0022-510X(00)00317-8
    [53] Milani P, Amadio M, Laforenza U, et al. (2013) Posttranscriptional regulation of SOD1 gene expression under oxidative stress: Potential role of ELAV proteins in sporadic ALS. Neurobiol Dis 60: 51-60.
    [54] Cereda C, Leoni E, Milani P, et al. (2013) Altered intracellular localization of SOD1 in leukocytes from patients with sporadic amyotrophic lateral sclerosis. PlOS One 8: e75916. doi: 10.1371/journal.pone.0075916
    [55] Smith RG, Henry YK, Mattson MP, et al. (1998) Presence of 4-hydroxynonenal in cerebrospinal fluid of patients with sporadic amyotrophic lateral sclerosis. Ann Neurol 44: 696-699. doi: 10.1002/ana.410440419
    [56] Ihara Y, Nobukuni K, Takata H, et al. (2005) Oxidative stress and metal content in blood and cerebrospinal fluid of amyotrophic lateral sclerosis patients with and without a Cu, Zn-superoxide dismutase mutation. Neurol Res 27: 105-108. doi: 10.1179/016164105X18430
    [57] Kirby J, Halligan E, Baptista MJ, et al. (2005) Mutant SOD1 alters the motor neuronal transcriptome: implications for familial ALS. Brain 128: 1686-1706.
    [58] Mimoto T, Miyazaki K, Morimoto N, et al. (2012) Impaired antioxydative Keap1/Nrf2 system and the downstream stress protein responses in the motor neuron of ALS model mice. Brain Res 1446: 109-118. doi: 10.1016/j.brainres.2011.12.064
    [59] Petri S, Korner S, Kiaei M (2012) Nrf2/ARE Signaling Pathway: Key Mediator in Oxidative Stress and Potential Therapeutic Target in ALS. Neurol Res Int: 878030.
    [60] Sarlette A, Krampfl K, Grothe C, et al. (2008) Nuclear erythroid 2-related factor 2-antioxidative response element signaling pathway in motor cortex and spinal cord in amyotrophic lateral sclerosis. J Neuropath Exp Neurol 67: 1055-1062. doi: 10.1097/NEN.0b013e31818b4906
    [61] Cao SS, Kaufman RJ (2014) Endoplasmic reticulum stress and oxidative stress in cell fate decision and human disease. Antioxid Redox Signaling 21: 396-413. doi: 10.1089/ars.2014.5851
    [62] Begum G, Harvey L, Dixon CE, et al. (2013) ER stress and effects of DHA as an ER stress inhibitor. Translational Stroke Res 4: 635-642. doi: 10.1007/s12975-013-0282-1
    [63] Bellucci A, Navarria L, Zaltieri M, et al. (2011) Induction of the unfolded protein response by alpha-synuclein in experimental models of Parkinson's disease. J Neurochem 116: 588-605. doi: 10.1111/j.1471-4159.2010.07143.x
    [64] Colla E, Jensen PH, Pletnikova O, et al. (2012) Accumulation of toxic alpha-synuclein oligomer within endoplasmic reticulum occurs in alpha-synucleinopathy in vivo. J Neurosci 32: 3301-3305. doi: 10.1523/JNEUROSCI.5368-11.2012
    [65] Nishitoh H, Kadowaki H, Nagai A, et al. (2008) ALS-linked mutant SOD1 induces ER stress- and ASK1-dependent motor neuron death by targeting Derlin-1. Genes Dev 22: 1451-1464. doi: 10.1101/gad.1640108
    [66] Atkin JD, Farg MA, Soo KY, et al. (2014) Mutant SOD1 inhibits ER-Golgi transport in amyotrophic lateral sclerosis. J Neurochem 129: 190-204. doi: 10.1111/jnc.12493
    [67] Hetz C, Mollereau B (2014) Disturbance of endoplasmic reticulum proteostasis in neurodegenerative diseases. Nat Rev Neurosci 15: 233-249.
    [68] Mercado G, Valdes P, Hetz C (2013) An ERcentric view of Parkinson's disease. Trends Mol Med 19: 165-175. doi: 10.1016/j.molmed.2012.12.005
    [69] Hoozemans JJ, van Haastert ES, Eikelenboom P, et al. (2007) Activation of the unfolded protein response in Parkinson's disease. Biochem Bioph Res Commun 354: 707-711. doi: 10.1016/j.bbrc.2007.01.043
    [70] Slodzinski H, Moran LB, Michael GJ, et al. (2009) Homocysteine-induced endoplasmic reticulum protein (herp) is up-regulated in parkinsonian substantia nigra and present in the core of Lewy bodies. Clin Neuropathol 28: 333-343.
    [71] Holtz WA, Turetzky JM, Jong YJ, et al. (2006) Oxidative stress-triggered unfolded protein response is upstream of intrinsic cell death evoked by parkinsonian mimetics. J Neurochem 99: 54-69. doi: 10.1111/j.1471-4159.2006.04025.x
    [72] Dukes AA, Van Laar VS, Cascio M, et al. (2008) Changes in endoplasmic reticulum stress proteins and aldolase A in cells exposed to dopamine. J Neurochem 106: 333-346. doi: 10.1111/j.1471-4159.2008.05392.x
    [73] Tinsley RB, Bye CR, Parish CL, et al. (2009) Dopamine D2 receptor knockout mice develop features of Parkinson disease. Ann Neurol 66: 472-484. doi: 10.1002/ana.21716
    [74] Mercado G, Castillo V, Soto P, et al. (2016) ER stress and Parkinson's disease: Pathological inputs that converge into the secretory pathway. Brain Res 1648: 626-632. doi: 10.1016/j.brainres.2016.04.042
    [75] Walker AK, Atkin JD (2011) Stress signaling from the endoplasmic reticulum: A central player in the pathogenesis of amyotrophic lateral sclerosis. IUBMB Life 63: 754-763.
    [76] Hetz C, Thielen P, Matus S, et al. (2009) XBP-1 deficiency in the nervous system protects against amyotrophic lateral sclerosis by increasing autophagy. Genes Dev 23: 2294-2306. doi: 10.1101/gad.1830709
    [77] Wang L, Popko B, Roos RP (2014) An enhanced integrated stress response ameliorates mutant SOD1-induced ALS. Hum Mol Genet 23: 2629-2638. doi: 10.1093/hmg/ddt658
    [78] Carreras-Sureda A, Pihan P, Hetz C (2017) The Unfolded Protein Response: At the Intersection between Endoplasmic Reticulum Function and Mitochondrial Bioenergetics. Front Oncol 7: 55.
    [79] Erpapazoglou Z, Mouton-Liger F, Corti O (2017) From dysfunctional endoplasmic reticulum-mitochondria coupling to neurodegeneration. Neurochem Int.
    [80] Eletto D, Chevet E, Argon Y, et al. (2014) Redox controls UPR to control redox. J Cell Sci 127: 3649-3658. doi: 10.1242/jcs.153643
    [81] Zhang K, Kaufman RJ (2008) From endoplasmic-reticulum stress to the inflammatory response. Nature 454: 455-462.
    [82] Tu BP, Weissman JS (2004) Oxidative protein folding in eukaryotes: mechanisms and consequences. J Cell Biol 164: 341-346. doi: 10.1083/jcb.200311055
    [83] Cuozzo JW, Kaiser CA (1999) Competition between glutathione and protein thiols for disulphide-bond formation. Nat Cell Biol 1: 130-135. doi: 10.1038/11047
    [84] Perri E, Parakh S, Atkin J (2017) Protein Disulphide Isomerases: emerging roles of PDI and ERp57 in the nervous system and as therapeutic targets for ALS. Exp Opin Targets 21: 37-49. doi: 10.1080/14728222.2016.1254197
    [85] Perri ER, Thomas CJ, Parakh S, et al. (2015) The Unfolded Protein Response and the Role of Protein Disulfide Isomerase in Neurodegeneration. Front Cell Dev Biol 3: 80.
    [86] Chaudhari N, Talwar P, Parimisetty A, et al. (2014) A molecular web: endoplasmic reticulum stress, inflammation, and oxidative stress. Front Cell Neurosci 8: 213.
    [87] Chiribau CB, Gaccioli F, Huang CC, et al. (2010) Molecular symbiosis of CHOP and C/EBP beta isoform LIP contributes to endoplasmic reticulum stress-induced apoptosis. Mol Cell Biol 30: 3722-3731. doi: 10.1128/MCB.01507-09
    [88] Yamaguchi H, Wang HG (2004) CHOP is involved in endoplasmic reticulum stress-induced apoptosis by enhancing DR5 expression in human carcinoma cells. J Biol Chem 279: 45495-45502. doi: 10.1074/jbc.M406933200
    [89] Lu M, Lawrence DA, Marsters S, et al. (2014) Cell death. Opposing unfolded-protein-response signals converge on death receptor 5 to control apoptosis. Science 345: 98-101.
    [90] Li G, Mongillo M, Chin KT, et al (2009) Role of ERO1-alpha-mediated stimulation of inositol 1,4,5-triphosphate receptor activity in endoplasmic reticulum stress-induced apoptosis. J Cell biol 186: 783-792.
    [91] Marciniak SJ, Yun CY, Oyadomari S, et al. (2004) CHOP induces death by promoting protein synthesis and oxidation in the stressed endoplasmic reticulum. Gene Dev 18: 3066-3077. doi: 10.1101/gad.1250704
    [92] Chen G, Bower KA, Ma C, et al. (2004) Glycogen synthase kinase 3beta (GSK3beta) mediates 6-hydroxydopamine-induced neuronal death. FASEB J 18: 1162-1164.
    [93] McNeill A, Magalhaes J, Shen C, et al. (2014) Ambroxol improves lysosomal biochemistry in glucocerebrosidase mutation-linked Parkinson disease cells. Brain 137: 1481-1495.
    [94] Prell T, Lautenschlager J, Weidemann L, et al. (2014) Endoplasmic reticulum stress is accompanied by activation of NF-kappaB in amyotrophic lateral sclerosis. J Neuroimmunol 270: 29-36. doi: 10.1016/j.jneuroim.2014.03.005
    [95] Yang W, Tiffany-Castiglioni E, Koh HC, et al. (2009) Paraquat activates the IRE1/ASK1/JNK cascade associated with apoptosis in human neuroblastoma SH-SY5Y cells. Toxicol Lett 191: 203-210. doi: 10.1016/j.toxlet.2009.08.024
    [96] Chang L, Karin M (2001) Mammalian MAP kinase signalling cascades. Nature 410: 37-40. doi: 10.1038/35065000
    [97] Darling NJ, Cook SJ (2014) The role of MAPK signalling pathways in the response to endoplasmic reticulum stress. BBA 1843: 2150-2163.
    [98] Davis RJ (2000): Signal transduction by the JNK group of MAP kinases. Cell 103: 239-252.
    [99] Abais JM, Xia M, Zhang Y, et al. (2014) Redox Regulation of NLRP3 Inflammasomes: ROS as Trigger or Effector? Antioxid Redox Signaling 22: 1111-1129.
    [100] Jope RS, Yuskaitis CJ, Beurel E (2007) Glycogen synthase kinase-3 (GSK3): inflammation, diseases, and therapeutics. Neurochem Res 32: 577-595. doi: 10.1007/s11064-006-9128-5
    [101] Nijholt DA, Nolle A, van Haastert ES, et al. (2013) Unfolded protein response activates glycogen synthase kinase-3 via selective lysosomal degradation. Neurobiol Aging 34: 1759-1771. doi: 10.1016/j.neurobiolaging.2013.01.008
    [102] Meares GP, Mines MA, Beurel E, et al. (2011) Glycogen synthase kinase-3 regulates endoplasmic reticulum (ER) stress-induced CHOP expression in neuronal cells. Exp Cell Res 317: 1621-1628. doi: 10.1016/j.yexcr.2011.02.012
    [103] Giordano S, Darley-Usmar V, Zhang J (2014) Autophagy as an essential cellular antioxidant pathway in neurodegenerative disease. Redox Biol 2: 82-90. doi: 10.1016/j.redox.2013.12.013
    [104] Loos B, Engelbrecht AM, Lockshin RA, et al. (2013) The variability of autophagy and cell death susceptibility: Unanswered questions. Autophagy 9: 1270-1285. doi: 10.4161/auto.25560
    [105] Scheper W, Nijholt DA, Hoozemans JJ (2011) The unfolded protein response and proteostasis in Alzheimer disease: preferential activation of autophagy by endoplasmic reticulum stress. Autophagy 7: 910-911. doi: 10.4161/auto.7.8.15761
    [106] Deegan S, Saveljeva S, Logue SE, et al. (2014) Deficiency in the mitochondrial apoptotic pathway reveals the toxic potential of autophagy under ER stress conditions. Autophagy 10: 1921-1936. doi: 10.4161/15548627.2014.981790
    [107] Madeo F, Eisenberg T, Kroemer G (2009) Autophagy for the avoidance of neurodegeneration. Gene Dev 23: 2253-2259. doi: 10.1101/gad.1858009
    [108] Cai Y, Arikkath J, Yang L, et al. (2016) Interplay of endoplasmic reticulum stress and autophagy in neurodegenerative disorders. Autophagy 12: 225-244.
    [109] Cortes CJ, Miranda HC, Frankowski H, et al. (2014) Polyglutamine-expanded androgen receptor interferes with TFEB to elicit autophagy defects in SBMA. Nat Neurosci 17: 1180-1189. doi: 10.1038/nn.3787
    [110] Palmieri M, Impey S, Kang H, et al. (2011) Characterization of the CLEAR network reveals an integrated control of cellular clearance pathways. Hum Mol Genet 20: 3852-3866. doi: 10.1093/hmg/ddr306
    [111] Brehme M, Voisine C, Rolland T, et al. (2014) A chaperome subnetwork safeguards proteostasis in aging and neurodegenerative disease. Cell Rep 9: 1135-1150. doi: 10.1016/j.celrep.2014.09.042
    [112] Genereux JC, Qu S, Zhou M, et al. (2014) Unfolded protein response-induced ERdj3 secretion links ER stress to extracellular proteostasis. EMBO J.
    [113] Montane J, Cadavez L, Novials A (2014) Stress and the inflammatory process: a major cause of pancreatic cell death in type 2 diabetes. Diabetes, metab syndrome obesity: targets ther 7: 25-34.
    [114] Song W, Wang F, Savini M, et al. (2013) TFEB regulates lysosomal proteostasis. Hum Mol Genet 22: 1994-2009.
    [115] Tan YL, Genereux JC, Pankow S, et al. (2014) ERdj3 is an endoplasmic reticulum degradation factor for mutant glucocerebrosidase variants linked to Gaucher's disease. Chem Biol 21: 967-976. doi: 10.1016/j.chembiol.2014.06.008
    [116] Wei H, Kim SJ, Zhang Z, et al. (2008) ER and oxidative stresses are common mediators of apoptosis in both neurodegenerative and non-neurodegenerative lysosomal storage disorders and are alleviated by chemical chaperones. Hum Mol Genet 17: 469-477.
    [117] Sybertz E, Krainc D (2014) Development of targeted therapies for Parkinson's disease and related synucleinopathies. J Lipid Res 55: 1996-2003. doi: 10.1194/jlr.R047381
    [118] Duplan E, Giaime E, Viotti J, et al. (2013) ER-stress-associated functional link between Parkin and DJ-1 via a transcriptional cascade involving the tumor suppressor p53 and the spliced X-box binding protein XBP-1. J Cell Sci 126: 2124-2133. doi: 10.1242/jcs.127340
    [119] Yokota T, Sugawara K, Ito K, et al. (2003) Down regulation of DJ-1 enhances cell death by oxidative stress, ER stress, and proteasome inhibition. Biochem Biophys Res Commun 312: 1342-1348. doi: 10.1016/j.bbrc.2003.11.056
    [120] Sajjad MU, Green EW, Miller-Fleming L, et al. (2014) DJ-1 modulates aggregation and pathogenesis in models of Huntington's disease. Hum Mol Genet 23: 755-766.
    [121] Shendelman S, Jonason A, Martinat C, et al. (2004) DJ-1 is a redox-dependent molecular chaperone that inhibits alpha-synuclein aggregate formation. PLOS Biol 2: e362. doi: 10.1371/journal.pbio.0020362
    [122] Jarvela TS, Lam HA, Helwig M, et al. (2016) The neural chaperone proSAAS blocks alpha-synuclein fibrillation and neurotoxicity. P Natl Acad Sci UAS 113: E4708-4715. doi: 10.1073/pnas.1601091113
    [123] Carra S, Rusmini P, Crippa V, et al. (2013) Different anti-aggregation and pro-degradative functions of the members of the mammalian sHSP family in neurological disorders. Phil Trans R Soc B 368: 20110409.
    [124] Chaari A, Hoarau-Vechot J, Ladjimi M (2013) Applying chaperones to protein-misfolding disorders: molecular chaperones against alpha-synuclein in Parkinson's disease. Int J Boil macromolecules 60: 196-205. doi: 10.1016/j.ijbiomac.2013.05.032
    [125] Fontaine SN, Martin MD, Dickey CA (2016) Neurodegeneration and the Heat Shock Protein 70 Machinery: Implications for Therapeutic Development. Curr Top Med Chem 16: 2741-2752. doi: 10.2174/1568026616666160413140741
    [126] Lindberg I, Shorter J, Wiseman RL (2015) Chaperones in Neurodegeneration. J Neurosci 35: 13853-13859. doi: 10.1523/JNEUROSCI.2600-15.2015
    [127] Chen S, Brown IR (2007) Neuronal expression of constitutive heat shock proteins: implications for neurodegenerative diseases. Cell Stress Chaperon 12: 51-58. doi: 10.1379/CSC-236R.1
    [128] Galbiati M, Crippa V, Rusmini P, et al. (2014) ALS-related misfolded protein management in motor neurons and muscle cells. Neurochem Int 79: 70-78. doi: 10.1016/j.neuint.2014.10.007
    [129] Papsdorf K, Richter K (2014) Protein folding, misfolding and quality control: the role of molecular chaperones. Essays Biochem 56: 53-68. doi: 10.1042/bse0560053
    [130] Baluchnejadmojarad T, Roghani M, Nadoushan MR, et al. (2009) Neuroprotective effect of genistein in 6-hydroxydopamine hemi-parkinsonian rat model. Phytother Res 23: 132-135. doi: 10.1002/ptr.2564
    [131] Choi BS, Kim H, Lee HJ, et al. (2014) Celastrol from 'Thunder God Vine' protects SH-SY5Y cells through the preservation of mitochondrial function and inhibition of p38 MAPK in a rotenone model of Parkinson's disease. Neurochem Res 39: 84-96. doi: 10.1007/s11064-013-1193-y
    [132] Inden M, Kitamura Y, Takeuchi H, et al. (2007) Neurodegeneration of mouse nigrostriatal dopaminergic system induced by repeated oral administration of rotenone is prevented by 4-phenylbutyrate, a chemical chaperone. J Neurochem 101: 1491-1504. doi: 10.1111/j.1471-4159.2006.04440.x
    [133] Jiang HQ, Ren M, Jiang HZ, et al. (2014) Guanabenz delays the onset of disease symptoms, extends lifespan, improves motor performance and attenuates motor neuron loss in the SOD1 G93A mouse model of amyotrophic lateral sclerosis. Neurosci 277: 132-138.
    [134] Mortiboys H, Aasly J, Bandmann O (2013) Ursocholanic acid rescues mitochondrial function in common forms of familial Parkinson's disease. Brain 136: 3038-3050.
    [135] Ono K, Ikemoto M, Kawarabayashi T, et al. (2009) A chemical chaperone, sodium 4-phenylbutyric acid, attenuates the pathogenic potency in human alpha-synuclein A30P + A53T transgenic mice. Parkinsonism Relat D 15: 649-654.
    [136] Ozsoy O, Seval-Celik Y, Hacioglu G, et al. (2011) The influence and the mechanism of docosahexaenoic acid on a mouse model of Parkinson's disease. Neurochem Int 59: 664-670. doi: 10.1016/j.neuint.2011.06.012
    [137] Richter F, Fleming SM, Watson M, et al. (2014) A GCase chaperone improves motor function in a mouse model of synucleinopathy. Neurotherapeutics 11: 840-856. doi: 10.1007/s13311-014-0294-x
    [138] Saxena S, Cabuy E, Caroni P (2009) A role for motoneuron subtype-selective ER stress in disease manifestations of FALS mice. Nature Neurosci 12: 627-636. doi: 10.1038/nn.2297
    [139] Kameta N, Masuda M, Shimizu T (2012) Soft nanotube hydrogels functioning as artificial chaperones. ACS Nano 6: 5249-5258. doi: 10.1021/nn301041y
    [140] Song W, Soo Lee S, Savini M, et al. (2014) Ceria nanoparticles stabilized by organic surface coatings activate the lysosome-autophagy system and enhance autophagic clearance. ACS Nano 8: 10328-10342. doi: 10.1021/nn505073u
    [141] Wang W, Sreekumar PG, Valluripalli V, et al. (2014) Protein polymer nanoparticles engineered as chaperones protect against apoptosis in human retinal pigment epithelial cells. J Controlled release 191: 4-14. doi: 10.1016/j.jconrel.2014.04.028
    [142] Liao YH, Chang YJ, Yoshiike Y, et al. (2012) Negatively charged gold nanoparticles inhibit Alzheimer's amyloid-beta fibrillization, induce fibril dissociation, and mitigate neurotoxicity. Small 8: 3631-3639. doi: 10.1002/smll.201201068
    [143] Palmal S, Maity AR, Singh BK, et al. (2014) Inhibition of amyloid fibril growth and dissolution of amyloid fibrils by curcumin-gold nanoparticles. Chemistry 20: 6184-6191. doi: 10.1002/chem.201400079
  • This article has been cited by:

    1. Lin Fan, Shunchu Li, Dongfeng Shao, Xueqian Fu, Pan Liu, Qinmin Gui, Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations, 2022, 7, 2473-6988, 11972, 10.3934/math.2022667
    2. Tingrong Jiang, Pengshe Zheng, Lin Xu, Lihui Leng, Application of elastic transformation method and similarity construction method in solving ordinary differential equations, 2023, 1598-5865, 10.1007/s12190-023-01958-5
    3. 雪倩 付, Solving a Class of First-Order Nonlinear Ordinary Differential Equations Which Can Be Transformed into Confluent Hypergeometric Equation Based on Elastic Upgrading Transformation Method, 2023, 13, 2160-7583, 1227, 10.12677/PM.2023.135126
    4. Peng E, Tingting Xu, Linhua Deng, Yulin Shan, Miao Wan, Weihong Zhou, Solutions of a class of higher order variable coefficient homogeneous differential equations, 2025, 20, 1556-1801, 213, 10.3934/nhm.2025011
    5. Jinfeng Liu, Pengshe Zheng, Jiajia Xie, Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction Methods, 2025, 5, 2673-9909, 41, 10.3390/appliedmath5020041
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6798) PDF downloads(1096) Cited by(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog