Uncertain Fractional Differential Equations Research Articles

Carbon option pricing based on uncertain fractional differential equation: A binomial tree approach

In this paper, we use a Caputo-Hadamard uncertain fractional differential equation (UFDE) to describe the changes of carbon emission rights price. Based on a binomial tree approach, a portfolio is constructed by buying options and shorting the underlying assets in a certain proportion. The pricing formulas for carbon finance European option are presented, and the algorithms are designed to formulate the price of carbon finance American option without explicit pricing formulas. Besides, a moment estimation method is used to obtain the parameter values in Caputo-Hadamard UFDE, and a method is provided to simulate the observation data of solution of Caputo-Hadamard UFDE. Finally, the specific Caputo-Hadamard UFDE with mean-reverting process is applied to simulating the change of carbon emission rights price in Chinese carbon market, and the feasibility of the proposed carbon option pricing method is illustrated by some numerical examples.

Least squares parameter estimation for uncertain fractional differential equations and application to stock model

In recent years, uncertain fractional differential equations was proposed for the description of complex uncertain dynamic systems with historical characteristics. For wider applications of uncertain fractional differential equations, researches on parameter estimation for uncertain fractional differential equations are of great importance. In this paper, based on the thought of least squares estimation and uncertain hypothesis test, an algorithm of parameter estimation for uncertain fractional differential equations is discussed. Finally, we consider the application of uncertain fractional differential equations based model to predict the forecasting stock price of three major indexes of U.S. stocks and make a comparison between uncertain fractional differential equations, uncertain differential equations and stochastic differential equations.

A new analytical algorithm for uncertain fractional differential equations in the fuzzy conformable sense

<abstract><p>This paper aims to explore and examine a fractional differential equation in the fuzzy conformable derivative sense. To achieve this goal, a novel analytical algorithm is formulated based on the Laplace-residual power series method to solve the fuzzy conformable fractional differential equations. The methodology being used to discover the fuzzy solutions depends on converting the desired equations into two fractional crisp systems expressed in $ \wp $-cut form. The main objective of our algorithm is to transform the systems into fuzzy conformable Laplace space. The transformation simplifies the system by reducing its order and turning it into an easy-to-solve algorithmic equation. The solutions of three important applications are provided in a fuzzy convergent conformable fractional series. Both the theoretical and numerical implications of the fuzzy conformable concept are explored about the consequential outcomes. The convergence analysis and theorems of the developed algorithm are also studied and analyzed in this regard. Additionally, this article showcases a selection of results through the use of both two-dimensional and three-dimensional graphs. Ultimately, the findings of this study underscore the efficacy, speed, and ease of the Laplace-residual power series algorithm in finding solutions for uncertain models that arise in various physical phenomena.</p></abstract>

Nonparametric estimation for uncertain fractional differential equations

After uncertainty theory was established, it has become a new branch of mathematics and been applied to describing the indeterministic phenomena as an uncertain dynamic system. As an important part of uncertainty theory, uncertain fractional differential equation is a good tool to model some complex dynamic systems with uncertainties. Due to the lack of corresponding information, models of parametric uncertain fractional differential equation are not always available. Therefore, the requirement for nonparametric estimation of uncertain fractional differential equation is urgent. Utilizing the thought of Legendre polynomial approximation, we propose a method to estimate the nonparametric uncertain fractional differential equations. Then, the error analysis of this method is given. After that, the numerical simulation is applied to illustrating the validity of error analysis and show the errors between the approximating functions and actual functions. The stability of this method has also been verified without highly dense observations. Finally, by means of uncertain hypothesis test and Kolmogorov–Smirnov test, we prove the superiority of uncertain fractional differential equations for some examples.

Extreme values of solution of Caputo–Hadamard uncertain fractional differential equation and applications

Uncertain fractional differential equation (UFDE) is a useful tool for studying complex systems in uncertain environments. The mathematical characteristics of solution of an UFDE are also widely used in various fields. In this paper, we give the extreme value theorems of solution of Caputo–Hadamard UFDE and applications. A numerical algorithm for obtaining the inverse uncertainty distributions (IUDs) for extreme values of solution of Caputo–Hadamard UFDE is presented; the stability and feasibility of the proposed algorithm are validated by numerical experiments. As an application of extreme value theorems in uncertain financial market, the pricing formulas of the American option based on the new uncertain stock model are given. Besides, the algorithms for computing the price of the American option without explicit pricing formulas based on the Simpson's rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments.

Reliability modelling with uncertain threshold and fractional differential degenerate equation in uncertain environment

A dependent competitive failure model of Caputo uncertain fractional differential equation (UFDE) degradation with uncertain threshold is proposed. There are two main reasons for the failure of an equipment or system, long-term natural wear and the impact of external loads. Each shock will cause a sharp increase in wear. In the absence of fault data, the natural wear process is driven by a Caputo type of UFDE, and the external load is an uncertain renewal reward process. Three different shock models are considered and calculated the belief reliability of device according to uncertainty theory, taking the jet pipe servo control valve as an example, the proposed model is simulated, and the results show that the proposed model is effective.

Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation

Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation

European option pricing problem based on a class of Caputo-Hadamard uncertain fractional differential equation

<abstract><p>Uncertain fractional differential equation (UFDE) is very suitable for describing the dynamic change in uncertain environments. In this paper, we consider the European option pricing problem by applying the Caputo-Hadamard UFDEs to simulate the dynamic change of stock price. First, an uncertain stock model with the mean-reverting process is studied, and the European option pricing formulas are given. Then, the effect of uncertain interference on the bond is considered, and the corresponding European option pricing formulas are presented. Finally, some numerical examples are given to illustrate the effectiveness of pricing formulas.</p></abstract>

A Class of Generalized Uncertain Fractional Differential Equations

In this paper, two generalized uncertain fractional differential equations (UFDEs) are proposed, which extend Riemann–Liouville type and Hadamard type into a uniform modality. Then the analytical solutions to several linear generalized (Caputo) UFDEs are provided. Furthermore, an existence and uniqueness theorem of solution for a generalized Caputo UFDE is proven.

An Approach for Numerical Solutions of Caputo–Hadamard Uncertain Fractional Differential Equations

This paper is devoted to investigating a numerical scheme for solving the Caputo–Hadamard uncertain fractional differential equations (UFDEs) arising from nonlinear uncertain dynamic systems. In our approach, we define an α-path, which is a link between a Caputo–Hadamard UFDE and a Caputo–Hadamard fractional differential equation and is the inverse uncertainty distribution of a Caputo–Hadamard UFDE. Then, a formula for calculating the expected value of the Caputo–Hadamard UFDE is studied. With the help of the modified predictor–corrector method, some numerical algorithms for the inverse uncertainty distribution and the expected value of the solution of Caputo–Hadamard UFDEs are designed. Corresponding numerical examples are given to confirm the validity and accuracy of the proposed algorithms.

Pricing Problem of Geometric Average Asian Barrier Option Based on Uncertain Fractional Differential Equation

In this paper, we investigate pricing problems of geometric average Asian barrier options based on a Caputo type of uncertain fractional differential equation. Unlike traditional stock models based on stochastic/ uncertain processes, we assume that underlying assets are subject to uncertain fractional processes. First, it provides a Caputo type of uncertain fractional mean-reverting model based on fractional differential equation. Then the pricing formulas are acquired for four geometric average Asian barrier options, namely, the up-and-in call option, the down-and-in put option, the up-and-out put option and the down-and-out call option. Meanwhile, numerical simulations are devised to validate the results of the model.

Reliability modeling of uncertain random fractional differential systems with competitive failures

Typical degradation-shock failure processes have been widely investigated in current researches, and the failures caused by their dependence are described as competitive failure processes. This paper explores competitive failure modes for uncertain random fractional systems involving degradation and shock processes. We develop a wear degradation model explicitly by employing uncertain fractional differential equations in order to demonstrate the potential heredity and memorability of a system. External shocks are then considered to follow a Poisson process. Based on the classification of shock types, three definitions of reliability index for competitive failures are presented. The reliability index formulas are derived for systems with extreme shock, cumulative shock, and δ shock using chance measures. Finally, we introduce a numerical example where the results of the reliability analysis confirm the validity of proposed reliability evaluation methods.

Uncertain Currency Option Pricing Based on the Fractional Differential Equation in the Caputo Sense

The foreign exchange market comprises the largest global volume, so the pricing of foreign exchange options has always been a hot issue in the foreign exchange market. This paper treats the exchange rate as an uncertain process that is described by an uncertain fractional differential equation, and establishes a new uncertain fractional currency model. The uncertain process is driven by Liu process, and, with the application of the Mittag-Leffler function, the solution of the fractional differential equation in a Caputo sense is presented. Then, according to the uncertain fractional currency model, the pricing formulas of European and American currency options are given. Lastly, the two numerical examples of European and American currency options are given; the price of the currency option increased when p changed from 1.0 to 1.1, and prices with different p were all decreasing functions of exercise price K.

Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations

Purpose: We construct the analytical approximate resiual power fuzzy series solutions of fuzzy conformable fractional differential equations in an -level depiction in the sense of strongly generalized -fuzzy conformable derivative in which of the all initial conditions are taken to be fuzzy numbers. Methodology: The certain fuzzy conformable fractional differential equation under strongly generalized -fuzzy derivative is converted to a crisp one as a family of differential inclusions and solved via resiual power method. The main drawback concerning the use of differential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a fuzzy valued function. Findings: (i) To show the efficiency of our proposed method: Several important and attractive test examples, which included the fractional conformable fuzzy integro-differential equation are discussed and solved in detail. (ii) To show the stability of approximate solutions to specific problems: some graphical results, numerical comparisons and tabulate data are created and discussed at different values of Value: Using the residual power series analysis methos is a powerful and easy-to-use analytic tool to solve initial problems on fuzzy conformable fractional differential equations and it successfully applied to solve real life problems such as the inductance–resistance–capacitance, RLC-series circuit.

Nonlinear impulsive problems for uncertain fractional differential equations

This paper deals with the impulsive problem for uncertain fractional dynamical system. Firstly, the concept of uncertain fractional impulsive problem involving the Caputo derivative is introduced and the analytic solutions to several linear uncertain fractional impulsive problems are derived with the help of the Mittag-Leffler functions. Then the existence and uniqueness theorems are developed via the Kuratowski measure of noncompactness and fixed point theorems, respectively. Finally, an illustrative example is provided to explain our main results.

Portfolio selection based on uncertain fractional differential equation

<abstract><p>Portfolio selection problems are considered in the paper. The securities in the proposed problems are suggested to follow uncertain fractional differential equations which have memory characteristics. By introducing the left semi-deviation of the wealth, two problems are proposed. One is to maximize the expected value and minimize the left semi-variance of the wealth. The other is to maximize the expected value of the wealth with a chance constraint that the left semi-deviation of the wealth is not less than a given number at a confidence level. The problems are equivalent to determinant ones which will be solved by genetic algorithm. Examples are provided to show the effectiveness of the proposed methods.</p></abstract>

Solutions and properties of Caputo-Erdelyi-Kober uncertain fractional differential equations

Solutions and properties of Caputo-Erdelyi-Kober uncertain fractional differential equations

New analysis of fuzzy fractional Langevin differential equations in Caputo's derivative sense

<abstract> <p>The extraction of analytical solution of uncertain fractional Langevin differential equations involving two independent fractional-order is frequently complex and difficult. As a result, developing a proper and comprehensive technique for the solution of this problem is very essential. In this article, we determine the explicit and analytical fuzzy solution for various classes of the fuzzy fractional Langevin differential equations (FFLDEs) with two independent fractional-orders both in homogeneous and non-homogeneous cases. The potential solution of FFLDEs is also extracted using the fuzzy Laplace transformation technique. Furthermore, the solution of FFLDEs is defined in terms of bivariate and trivariate Mittag-Leffler functions both in the general and special forms. FFLDEs are a new topic having many applications in science and engineering then to grasp the novelty of this work, we connect FFLDEs with RLC electrical circuit to visualize and support the theoretical results.</p></abstract>

Existence and uniqueness of solutions to uncertain fractional switched systems with an uncertain stock model

In this paper, an uncertain fractional switched system is a fractional switched system disturbed by subjective uncertainty, which can be written by Caputo type of uncertain fractional differential equations. Few results concerning uncertain fractional systems were published before. To fill this gap, the property of solutions to uncertain fractional switched systems with finite-time horizon is investigated in depth. Based on two conditions called linear growth condition and Lipschitz condition, an existence and uniqueness theorem of solutions is proposed for the uncertain fractional switched systems, and the strict demonstration is given for the theorem in terms of uncertainty theory and Banach fixed point theorem. Finally, an uncertain stock model is proposed and analyzed to illustrate the effectiveness of the results obtained.

Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market

Uncertain fractional differential equations (UFDEs) have non-locality features to reflect memory and hereditary characteristics for the asset price changes, thus are more suitable to model the real financial market. Based on this characteristic, this paper primarily investigates the monotonicity theorem for uncertain fractional differential equations in Caputo sense and its application. Firstly, monotonicity theorems for solutions of UFDEs are presented by using the α-path method. Secondly, as the application of the monotone function theorem, a novel uncertain fractional mean-reverting model with a floating interest rate is presented. Lastly, the pricing formulas of the European and American options are derived for the proposed model based on the monotone function and present extreme values and time integral theorems, respectively. In addition, numerical schemes are designed, and numerical calculations are illustrated concerning different parameters through the predictor–corrector method.