Fuzzy Stochastic Differential Equations Research Articles

Numerical solution of fuzzy stochastic Volterra integral equations with constant delay

This paper considers the nonlinear fuzzy stochastic Volterra integral equations with constant delay, which are general and include many fuzzy stochastic integral and differential equations discussed in literature. Since Doob's martingale inequality is no longer applicable to such equations, a new maximum inequality is obtained. Combining with the Picard approximation method, the existence and uniqueness of solutions to nonlinear fuzzy stochastic Volterra integral equations with constant delay are given. Moreover we prove that the solution behaves stably in the case of small changes of initial values, kernels and nonlinearities. We further develop a Euler-Maruyama (EM) scheme and prove the strong convergence of the scheme. It is shown that the strong convergence order of the EM method is 0.5 under Lipschitz condition. Moreover, the strong superconvergence order is 1 if further, the kernel h(t,s) of the stochastic term satisfies h(t,t)=0. Numerical examples demonstrated that the numerical results are consistent with the theoretical research conclusions. Furthermore, the application model of the fuzzy stochastic Volterra integral equation with constant delay in population dynamics is considered, and the exact solution of the numerical example is given in explicit form.

Approximate solutions for fuzzy stochastic differential equations with Markovian switching

The fuzzy stochastic differential equations with Markovian switching are considered. First, under the Carath e ´ odory assumptions, the existence and uniqueness theorem for the aforementioned equations is given by means of stopping time techniques, discretization method, and the Gronwall-Bellman inequality. Then the boundedness of the second moment of the solutions for such equations is established. Furthermore, by using the averaging method and stochastic analysis, as well as the Bihari's inequality, this paper mainly demonstrates the asymptotic relationship between the solutions of the original equations and the corresponding averaged equations in terms of mean square and probability. Finally, the theoretical result of the averaging method is illustrated by an example.

Existence and Uniqueness Theorem of Fuzzy Stochastic Ordinary Differential Equations

A fuzzy valued diffusion term, which in a fuzzy stochastic differential equation refers to one-dimensional Brownian motion, is defined by the meaning of the stochastic integral of a fuzzy process. In this paper, the existence and uniqueness theorem of fuzzy stochastic ordinary differential equations, based on the mean square convergence of the mathematical induction approximations to the associated stochastic integral equation, are stated and demonstrated.

Approximation solution of backward doubly fuzzy stochastic differential equations

In this paper, we propose a new formula for the backward doubly fuzzy stochastic differential equations (BFDSDEs), In the beginning, we present some basic concepts, definitions, and Hypotheses to obtain the numerical scheme for BFDSDEs, as our scheme depends on the partition of interval [0, T]. In our work, we prove that under Lipschitz conditions, the approximation solution for the backward fuzzy doubly stochastic differential equations converges to the exact solution by using mean square error, and prove the existence and uniqueness of approximations solutions to BFDSDEs.

Symmetric Fuzzy Stochastic Differential Equations Driven by Fractional Brownian Motion

We consider symmetric fuzzy stochastic differential equations where diffusion and drift terms arise in a symmetric way at both sides of the equations and diffusion parts are driven by fractional Brownian motions. Such equations can be used in real-life hybrid systems, which include properties of being both random and fuzzy and reflecting long-range dependence. By imposing on the mappings occurring in the equation the conditions of Lipschitzian continuity and additional constraints by an integrable stochastic process, we construct an approximation sequence of fuzzy stochastic processes and apply this to prove the existence of a unique solution to the studied equation. Finally, a model from population dynamics is considered to illustrate the potential application of our equations.

Ulam–Hyers stability of Caputo-type fractional fuzzy stochastic differential equations with delay

We explore a new kind of Caputo-type fractional fuzzy stochastic differential equations (FFSDEs) with delay. We establish the existence result of FFSDEs with delay by monotone iterative technique combined with the method of upper and lower solutions, and then the uniqueness is proved. Subsequently, we study the Ulam–Hyers (U–H) stability. Finally, three examples with numerical simulations are provided to illustrate the theoretical results.

An approximate approach to fuzzy stochastic differential equations under sub-fractional Brownian motion

In this paper, we introduce fuzzy stochastic differential equations (FSDEs) driven by sub-fractional Brownian motion (SFBM) which are applied to describe phenomena subjected to randomness and fuzziness simultaneously. The SFBM is an extension of the Brownian motion that retains many properties of fractional Brownian motion (FBM), but not the stationary increments. This property makes SFBM a possible candidate for models that include long-range dependence, self-similarity, and non-stationary increments which is suitable for the construction of stochastic models in finance and non-stationary queueing systems. We apply an approximation method to stochastic integrals, and a decomposition of the SFBM to find the existence and uniqueness of the solutions.

Application of fuzzy Malliavin calculus in hedging fixed strike lookback option

<abstract><p>In this paper, we develop a Malliavin calculus approach for hedging a fixed strike lookback option in fuzzy space. Due to the uncertainty in financial markets, it is not accurate to describe the problems of option pricing and hedging in terms of randomness alone. We consider a fuzzy pricing model by introducing a fuzzy stochastic differential equation with Skorohod sense. In this way, our model simultaneously involves randomness and fuzziness. A well-known hedging strategy for vanilla options is so-called $ \Delta $-hedging, which is usually derived from the Itô formula and some properties of partial differentiable equations. However, when dealing with some complex path-dependent options (such as lookback options), the major challenge is that the payoff function of these options may not be smooth, resulting in the estimates are computationally expensive. With the help of the Malliavin derivative and the Clark-Ocone formula, the difficulty will be readily solved, and it is also possible to apply this hedging strategy to fuzzy space. To obtain the explicit expression of the fuzzy hedging portfolio for lookback options, we adopt the Esscher transform and reflection principle techniques, which are beneficial to the calculation of the conditional expectation of fuzzy random variables and the payoff function with extremum, respectively. Some numerical examples are performed to analyze the sensitivity of the fuzzy hedging portfolio concerning model parameters and give the permissible range of the expected hedging portfolio of lookback options with uncertainty by a financial investor's subjective judgment.</p></abstract>

Averaging Principle for Fuzzy Stochastic Differential Equations

Averaging Principle for Fuzzy Stochastic Differential Equations

RESEARCH ON STOCHASTIC FUZZY DIFFERENTIAL EQUATIONS IN MULTIPLE BLURRED IMAGE REPAIR MODELS

This paper aims to study the processing and repairing methods of blurred images, promote the development of partial differential equations in the field of image processing, and expand the application of stochastic fuzzy differential equations in the field of image processing and repair. This study starts with a typical blurred image repair method. First, a comparative analysis of several common blurred image repair methods including Wiener filtering restoration, inverse filtering restoration, and Lucy–Richardson (L-R) filtering restoration is performed. Second, based on the linear partial differential equation learning model (LPDE), the concept of fuzzy integral is introduced, and an improved stochastic fuzzy partial differential equation learning model (SFCPDE) is proposed. The effect of the learning model before and after improvement on blurred color image processing is compared and analyzed. Finally, based on the total variation (TV) blurred image repair algorithm, an improved TV blurred image repair algorithm is proposed. The comparison and analysis of the repair effects of several blurred image repair algorithms are performed. The results show that there are obvious differences in the repairing methods with or without noise. Inverse filtering works best when there is no noise. L-R filtering has the disadvantage of amplifying noise. Compared with LDPE, the training speed of SFCPDE is significantly improved, and the training error is less than LDPE. The SFCPDE learning model performs better in the processing of blurred color images. After 10 iterations, the improved TV algorithm is significantly better than the TV algorithm and the CDD algorithm in repairing blurred images. The PSNR value of the TV algorithm and the curvature-driven diffusion (CDD) algorithm after 10 iterations corresponds to about 60% of the PSNR value of the improved algorithm. The algorithms and models of stochastic fuzzy partial differential equations proposed in this paper have great application potential in the processing and repair of multiple blurred images.

Fuzzy Stochastic Differential Equations Driven by a Fuzzy Brownian Motion

In our previous work, we study fuzzy Itô integrals driven by a fuzzy Brownian motion. In this article, we continue this study. The purpose of this paper is to study the weak uniqueness of fuzzy stochastic differential equations taking into account fuzzy Brownian motion. For instance, we construct the fuzzy stochastic differential equation driven by a fuzzy Brownian motion. To define and prove our results, we use the fuzzification, the alpha cut method and the Hausdorff distance between two fuzzy quantities. Some results are to our credit in this article like the instance, we construct the fuzzy stochastic differrential equation driven by fuzzy Brownian motion. Furthermore, we develop fuzzy Itô calculus driven by a fuzzy Brownian motion. Our result complement existing ones in that the fuzzy version of Brownian motion is taken into account.

Sensitivity of option prices via fuzzy Malliavin calculus

In this paper, we define the Malliavin derivative and the Skorohod integral for fuzzy stochastic processes for the sensitivity analysis of the option prices. The price dynamics could be modelled by equations involving uncertainties that lead to modelling with fuzzy processes in equations. We introduce a fuzzy stochastic differential equation for financial pricing models under an asset price that follows a fuzzy stochastic process. If there is no explicit solution to the equations, the sensitivity analysis, and the estimates are computationally expensive, and the result will be inaccurate due to the estimates and the derivative of the payoff function. The Malliavin calculus and the integration by parts formula in fuzzy space can be exploited to bypass the derivative of the payoff function.

Iterative Method for Non-Adapted Fuzzy Stochastic Differential Equations

Iterative Method for Non-Adapted Fuzzy Stochastic Differential Equations

Fuzzy stochastic differential equations driven by fractional Brownian motion

In this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.

Итерационный метод для неадаптированных нечетких стохастических дифференциальных уравнений

Итерационный метод для неадаптированных нечетких стохастических дифференциальных уравнений

FRACTAL DIMENSION OF FRACTIONAL BROWNIAN MOTION BASED ON RANDOM SETS

The fractal dimension of fractional Brownian motion can effectively describe random sets, reflecting the regularity implicit in complex random sets. Data mining algorithms based on fractal theory usually follow the calculation of the fractal dimension of fractional Brownian motion. However, the existing fractal dimension calculation methods of fractal Brownian motion have high time complexity and space complexity, which greatly reduces the efficiency of the algorithm and makes it difficult for the algorithm to adapt to high-speed and massive data flow environments. Therefore, several existing fractal dimension calculation methods of fractional Brownian motion are summarized and analyzed, and a random method is proposed, which uses a fixed memory space to quickly estimate the associated dimension of the data stream. Finally, a comparison experiment with existing algorithms proves the effectiveness of this random algorithm. Second, in the sense of two different measures, based on the principle of stochastic comparison, the stability of the stochastic fuzzy differential equations is derived using the stability of the comparison equations, and the practical stability criterion of two measures according to probability is obtained. Then, the stochastic fuzzy differential equations are discussed. The definition of stochastic exponential stability is given and the stochastic exponential stability criterion is proved.

Symmetric Fuzzy Stochastic Differential Equations with Generalized Global Lipschitz Condition

The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations. We claim that such symmetric equations have unique solutions in the case that equations’ coefficients satisfy a certain generalized Lipschitz condition. To show this, we prove that an approximation sequence converges to the solution. Then, a study on stability of solution is given. Some inferences for symmetric set-valued stochastic differential equations end the paper.

Optimal Release Time Determination Using FMOCCP Involving Randomized Cost Budget for FSDE-Based Software Reliability Growth Model

This paper presents a software reliability growth model (SRGM) based on a fuzzy stochastic differential equation (FSDE) approach. The software shall comprise of dependent and independent main and sub-components owing to a wide range of functionalities in the system. In order to obtain the optimal date of product release, a two-phase fuzzy goal programing (FGP) method has been applied for solving a fuzzy multi-objective chance-constrained programming (FMOCCP) model. While framing the cost function, the costs associated with changes in software specifications have been considered along with other cost factors. In most cases, the changes in software specifications during the software development process are dynamic in nature necessitating the cost budget to be considered as a random variable while formulating the fuzzy chance-constrained model. The SRGM proposed has been validated on real-time data sets and numerical illustrations have been provided to instantiate the optimal time of product release.

Fuzzy Malliavin derivative and linear Skorohod fuzzy stochastic differential equation

Fuzzy Malliavin derivative and linear Skorohod fuzzy stochastic differential equation

Stochastic differential equations with imprecisely defined parameters in market analysis

Risk and uncertainties plays a major role in stock market investments. It is a pedagogical practice to deduce probability distributions for analysing stock market returns using theoretical models of investor behaviour. Generally, economists estimate probability distributions for stock market returns that are observed from the history of past returns. Besides this, there are impreciseness involved in various factors affecting market investment and returns. As such, we need to model a more reliable strategy that will quantify the uncertainty with better confidence. Here, we have presented a computational method to solve fuzzy stochastic Volterra–Fredholm integral equation which is based on the block pulse functions (BPFs) using fuzzy stochastic operational matrix (SOM). The concept of fuzziness has been hybridized with BPFs, and the corresponding stochastic integral equation has been modelled. For illustration, the developed model has been used to investigate an example problem of Black–Scholes fuzzy stochastic differential equation (FSDE), and the results are compared in special cases.