Uncertain stock model with periodic dividends

Geometric Asian barrier option pricing formulas of uncertain stock model

The Liu process is a new tool to deal with the noise process based on uncertainty theory. In this paper, we view the foreign exchange rate as an uncertain processes, described by uncertain differential equations driven by the Liu process, and build an uncertain currency model. Then, the uncertain currency option problems are discussed. Moreover, European and American currency option pricing formulas are derived for the proposed uncertain currency model and some mathematical properties are studied. Finally, two numerical examples are documented.

In response to the potential failure of traditional models when faced with issues of nonwhite noise residuals and imprecise data, this study extends the mixed data sampling (MIDAS) model to the field of uncertainty theory to tackle these challenges. Under the framework of uncertainty theory, this research addresses the frequency inconsistency in economic data collection by constructing two types of uncertain MIDAS models, aiming to fill the gap in uncertainty theory's handling of predictive analysis for variables with different frequencies. Furthermore, this study integrates the dual perspectives of energy consumption and energy‐related carbon dioxide emissions. By building uncertain MIDAS models with uncertain disturbance terms and traditional MIDAS models, the research systematically establishes and comparatively analyzes univariate and multivariate energy consumption, departmental carbon dioxide emissions, and multivariate models that combine these two perspectives. The study's results not only confirm the nonwhite noise characteristics of residuals and validate the rationality of treating residuals as uncertain disturbance terms but also demonstrate through comparative analysis that the uncertain MIDAS model outperforms the traditional MIDAS model in terms of forecasting effectiveness. Moreover, the multivariate forecasting method that considers both perspectives can more comprehensively describe and predict the US quarterly gross domestic product (GDP), showing its superior predictive capability. Furthermore, by altering the evaluation criterion, substituting GDP with nominal GDP and introducing the control variables for robustness analysis, we have further verified the robustness of the model and its results.

This paper introduces an uncertain mean‐reverting currency model that incorporates floating domestic and foreign interest rates along with an exponential Ornstein–Uhlenbeck exchange rate process, all grounded in uncertainty theory. Pricing formulas for both Asian call and put options are derived within this framework. The parameters of the model are estimated using real financial data from Canada and the United States, including the Canadian Overnight Repo Rate Average (CORRA), the American Federal Funds Effective Rate (AFFER), and the monthly average exchange rate of the US Dollar to the Canadian Dollar (USDCAD). The method of moments is applied to estimate the unknown parameters, and goodness‐of‐fit tests are conducted to validate the parameter estimates. Numerical experiments demonstrate that Asian option prices decrease as domestic and foreign initial interest rates increase. The prices of call and put options show divergent behaviors with respect to the initial exchange rate and the fixed strike price. Additionally, the paper investigates the nonlinear relationship between option prices and expiration time. In the appendix, the uncertain currency model is transformed into a stochastic currency model, and statistical tests confirm its inapplicability to the selected data, thereby substantiating the choice of the uncertain currency model.

A lookback option is a path-dependent option, offering a payoff that depends on the maximum or minimum value of the underlying asset price over the life of the option. This paper presents a new mean-reverting uncertain stock model with a floating interest rate to study the lookback option price, in which the processing of the interest rate is assumed to be the uncertain counterpart of the Cox–Ingersoll–Ross (CIR) model. The CIR model can reflect the fluctuations in the interest rate and ensure that such rate is positive. Subsequently, lookback option pricing formulas are derived through the α-path method and some mathematical properties of the uncertain option pricing formulas are discussed. In addition, several numerical examples are given to illustrate the effectiveness of the proposed model.

Nowadays, one of the main challenges of marine maintenance is how to select an optimum maintenance strategy for each component of the complex ship machinery system. The uncertainty of the parameters is one of the main difficulties encountered. For example, engineers and experts are always questioning the credibility and integrity of the data collected, and historical maintenance records may also be lost during maintenance. The above data asymmetry will also lead to parameter uncertainty, which directly affects the accuracy of the prediction results. This paper proposes a method for determining maintenance types of ship equipment based on uncertainty theory and the Data Envelopment Analysis (DEA) model. Firstly, this paper constructs an uncertain maintenance optimization model (UMOM) based on the classic Data Envelopment Analysis model. Then, we converted the UMOM into an equivalent deterministic model for easy calculation by the uncertainty theory. Finally, a case study is given to verify this model. The results will conclude that the UMOM can meet the need for a reasonable classification of maintenance types of mechanical equipment in a marine system and provide valuable information for systemic management and storage.

Based on the theory of uncertain programming, in this paper, we establish an uncertain programming model for the emergency shipping route transportation. Considering the confidence level of the constraint function, the uncertain programming model is transformed into a certain model for the programming solution. A numerical example is used to verify the feasibility and effectiveness of the proposed model. The results of numerical examples show that the model and the algorithm are effective for solving multi-objective path optimization problems under uncertain conditions.KeywordsUncertainty theoryEmergency maritime routeUncertain programming

Enhanced index tracking (EIT) problem is concerned with selecting a tracking portfolio to beat the benchmark on return while having the minimum tracking error. This paper addresses the EIT problem based on uncertainty theory where stock returns are treated as uncertain variables instead of random variables. Under the framework of uncertainty theory, the paper proposes a new uncertain EIT model where the higher-order moment of the downside is used as the tracking error measure, as higher-order moment makes the model more widely applicable and the downside risk is in line with investors’ perception of risk. Besides, some realistic constraints are considered in the new uncertain EIT model. Then, the properties of the proposed model are discussed. To solve the model, we proposed, which is a nonlinear integer programming problem, a meta-heuristic algorithm presented. The efficiency of the algorithm and the applications of the proposed model are illustrated through numerical experiments.

A fixed charge transportation problem with damageable items under uncertain environment

In this paper, a multi-objective faculty course allocation problem with result analysis and feedback analysis based on uncertain preferences mathematical model is presented. To deal with an uncertain model, three different ranking criteria are being used to develop: a) Expected value, b) Optimistic value, c) Dependent optimistic value criterion. These mathematical models are transformed into their corresponding deterministic forms using the basic concepts of uncertainty theory. The deterministic model of DOCM consists of fractional objectives which are converted into their linear form using Charnes and Cooper’s transformation. These deterministic formulations MOFCAP are converted into a single objective problem by using the fuzzy programming technique with linear and exponential membership functions. Further, the single objective problem for all the defined models is solved in the Lingo 18.0 software to derive the Pareto-optimal solution. The sensitivity of the models is also performed to examine the variation in the objective function due to the variation in parameters. Finally, a numerical example is given to exhibit the application and algorithm of the models.

Stock model is used to describe the evolution of stock price in the form of differential equations. In early years, the stock price was assumed to follow a stochastic differential equation driven by a Brownian motion, and some famous models such as Black---Scholes stock model and Black---Karasinski stock model were widely used. This paper assumes that the stock price follows an uncertain differential equation driven by Liu process rather than Brownian motion, and accepts Liu's stock model to simulate the uncertain market. Then this paper proves a no-arbitrage determinant theorem for Liu's stock model and presents a sufficient and necessary condition for no-arbitrage. Finally, some examples are given to illustrate the usefulness of the no-arbitrage determinant theorem.

This paper copes with a joint Location-Allocation-Inventory problem in a three-echelon base-level spare part support system with epistemic uncertainty in uncertain demands of bases. The aim of the paper is to propose an optimization model under the uncertainty theory to minimize the total cost, which integrates crucial characterizations of the inventory control decisions and the location-allocation scheme arrangement under a periodic review order-up-to-S (T, S) policy. Uncertainty theory is introduced in this paper to characterize epistemic uncertainty, where demands are treated as uncertain variables and stockout loss is represented by value-at-risk in uncertain measurement. To solve the original uncertain optimization model, an equivalent deterministic model is derived and addressed by an improved bilevel genetic algorithm. Moreover, the proposed models and algorithm are encoded into numerical examples for supply chain programming. The results highlight the applicability of the model and the algorithm’s effectiveness in approaching the optimal solution compared with traditional genetic algorithm. Sensitivity analyses are further made for the impacts of review time and inventory capacity on different cost components.

Based on the current background of scenic spot management and tourism planning in China, this paper establishes an uncertain tourism route expectation-constrainted programming model under the framework of uncertainty theory. Finally, eight attractions in Jinan are used as practical cases to illustrate the usefulness of the model. The research results show that the proposed model can effectively alleviate the problems of unreasonable allocation of tourists' time and cost in the traditional model, and make the layout and programming of routes more reasonable and efficient.

In system reliability problems, the lifetimes and the repair times are usually indeterministic. When insufficient samples are available to estimate distribution functions of the lifetimes and the repair times, it is rational to invite some experts to evaluate the belief degrees. Uncertainty theory is a branch of mathematics for modelling belief degrees. In this paper, the lifetimes and the repair times of typical repairable systems are assumed to be uncertain variables. Based on the above-mentioned assumptions, the uncertain reliability mathematical models of simple repairable series systems, simple repairable parallel systems, simple repairable series-parallel systems, and simple repairable parallel–series systems are established, respectively. The reliability indices, such as steady state availability, steady state failure frequency, mean up time, and mean down time, are proposed. Finally, some numerical examples are given for illustration.

An uncertain single machine scheduling problem with periodic maintenance