Search

In this paper, we consider a compartmental model to fit the real data of confirmed active cases with COVID-19 in Portugal, from March 2, 2020 until September 10, 2021 in the Primary Care Cluster in Aveiro region, ACES BV, reported to the Public Health Unit. The model includes a deterministic component based on ordinary differential equations and a stochastic component based on bootstrap methods in regression. The main goal of this work is to take into account the variability underlying the data set and analyse the estimation accuracy of the model using a residual bootstrapped approach in order to compute confidence intervals for the prediction of COVID-19 confirmed active cases. All numerical simulations are performed in R environment (R version. 4.0.5). The proposed algorithm can be used, after a suitable adaptation, in other communicable diseases and outbreaks. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability of certain class of impulsive evolution stochastic functional differential equations, with variable delays, driven by a fractional Brownian motion in a separable real Hilbert space. We derive a new set of sufficient conditions for approximate controllability using a stochastic analysis of fractional Brownian motion with Hurst parameter H ∈ (1/2,1) and a Schaefer's fixed point theorem. An example is considered at the end of the paper to illustrate the obtained abstract results. [ABSTRACT FROM AUTHOR]

In this paper we further develop a notion of causal predictability defined in [A. Merkle, Predictability and Uniqueness of Weak Solutions of Stochastic Differential Equations, Analele Stiintifice ale Universitatii Ovidius Constanta, 2022] as a concept of dependence which is based on Granger's definition of causality. More precisely, in [A. Merkle, Predictability and Uniqueness of Weak Solutions of Stochastic Differential Equations, Analele Stiintifice ale Universitatii Ovidius Constanta, 2022] causal predictability is defined between filtrations, but now we introduce causal predictability between stochastic processes and filtrations. Also, we provide some properties of this new concept. Then we apply the given causality concept to the uniqueness of weak solutions of the stochastic differential equations and in financial mathematics. Granger [Investigating causal relations by econometric models and cross spectral methods, Econometrica. 37 (1969), pp. 424–438] has considered causality concept between time series. In this paper we consider continuous time processes, since continuous time models represent the first step in various applications, such as in finance, econometric practice, neuroscience, epidemiology, climatology, demographic, etc. [ABSTRACT FROM AUTHOR]

A formal approach to rephrasing nonlinear filtering of stochastic differential equations is the Kushner setting in applied mathematics and dynamical systems. Thanks to the ability of the Carleman linearization, the 'nonlinear' stochastic differential equation can be equivalently expressed as a finite system of 'bilinear' stochastic differential equations with the augmented state under the finite closure. Interestingly, the novelty of this paper is to embed the Carleman linearization into a stochastic evolution of the Markov process. The nonlinear swing equation is the cornerstone and lays the foundation of the power systems dynamics. To illustrate the Carleman linearization of the Markov process, this paper embeds the Carleman linearization into a nonlinear swing stochastic differential equation. Furthermore, we achieve the nonlinear swing equation filtering in the Carleman setting. Filtering in the Carleman setting has simplified algorithmic procedures. The concerning augmented state accounts for the nonlinearity as well as stochasticity. We show that filtering the nonlinear stochastic swing equation in the Carleman framework is more refined as well as sharper in contrast to the benchmark nonlinear extended Kalman filter (EKF). The Carleman filtering framework reduces approximately three times more conditional mean absolute errors as compared to the benchmark EKF method for the application of machine swing dynamics. This paper suggests the usefulness of the Carleman embedding into the stochastic differential equation to filter the concerning nonlinear stochastic differential system. This paper will interest nonlinear stochastic dynamists exploring and unfolding linearization embedding techniques to their research. [ABSTRACT FROM AUTHOR]

In a recent paper, Lanconelli and Scorolli10 extended to the multidimensional case a Wong–Zakai-type approximation for Itô stochastic differential equations (SDEs) proposed by Øksendal and Hu7. The aim of this paper is to extend the latter result to system of SDEs of Itô type driven by fractional Brownian motion (fBm) like those considered by Hu6. This extension is not trivial since the covariance structure of the fBm precludes us from using the same approach as that used by Lanconelli and Scorolli. Instead we employ a truncated Cameron–Martin expansion as the approximation for the fBm. We are naturally led to the investigation of a semilinear hyperbolic system of evolution equations in several space variables that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the number of terms in the expansion goes to infinite. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we focus on fractional stochastic differential equations (FSDEs) with a stochastic forcing term, i.e., to FSDE, we add a stochastic forcing term. Using the implicit scheme of Euler's approximation, the conditions for the existence and uniqueness of the solution of FSDEs with a stochastic forcing term are established. Such equations can be applied to considering FSDEs with a permeable wall. [ABSTRACT FROM AUTHOR]

While scanning or shooting a document, factors like ink density and paper transparency may cause the content from the reverse side to become visible through the paper, resulting in a digital image with a 'seen-through' phenomenon, which will affect practical applications. In addition, document images can be affected by random factors during the imaging process, such as differences in the performance of camera equipment and variations in the physical properties of the document itself. These random factors increase the noise of the document image and may cause the seen-through phenomena to become more complex and diverse. To tackle this issue, we propose the Fuzzy Diffusion Model (FDM), which combines fuzzy logic with diffusion models. It effectively models complex seen-through effects and handles uncertainties in document images. Specifically, we gradually degrade the original image with mean-reverting stochastic differential equation(SDE) to transform it into seen-through mean state with fixed Gaussian noise version. Following this, fuzzy operations are introduced into the noise network. Which helps the model better learn noise and data distributions by reasoning about the affiliation relationship of each pixel point through fuzzy logic. Eventually, in the reverse process, the low-quality image is gradually restored by simulating the corresponding reverse-time SDE. Extensive quantitative and qualitative experiments conducted on various datasets demonstrate that the proposed method significantly removes the seen-through effects and achieves good results under several metrics. The proposed FDM effectively solves the seen-through effects of document images and obtains better visual quality. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the global approximation of solutions of stochastic differential equations (SDEs) driven by countably dimensional Wiener process. Under certain regularity conditions imposed on the coefficients, we show lower bounds for exact asymptotic error behaviour. For that reason, we analyse separately two classes of admissible algorithms: based on equidistant, and possibly not equidistant meshes. Our results indicate that in both cases, decrease of any method error requires significant increase of the cost term, which is illustrated by the product of cost and error diverging to infinity. This is, however, not visible in the finite-dimensional case. In addition, we propose an implementable, path-independent Euler algorithm with adaptive step-size control, which is asymptotically optimal among algorithms using specified truncation levels of the underlying Wiener process. Our theoretical findings are supported by numerical simulation in Python language. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for H>12$$ \mathcal{H}>\frac{1}{2} $$ these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for H(ξ)≠12$$ \mathcal{H}\left(\xi \right)\ne \frac{1}{2} $$. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm. [ABSTRACT FROM AUTHOR]

This paper is concerned with the linear delayed forward–backward stochastic differential equations (D-FBSDEs). The existence of time delay leads to the infinite-dimensional problem which makes the explicit solvability more challenging. The main contribution is to propose a discretisation approach which gives the explicit solution of the D-FBSDEs and consists of the following three steps: first, we transform the continuous-time D-FBSDEs into the discrete-time form with the aid of interval partition. Second, we derive the solution of the discrete-time D-FBSDEs by applying backward iterative induction. Finally the explicit solution of the continuous-time D-FBSDEs is obtained by taking the limit to the solution of discrete-time form which is also strictly proved under continuous-time framework. The proposed approach is applicable for more general FBSDEs with delay, which would provide a complete solution to the stochastic LQ control with time delay. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a reflected backward stochastic differential equation (RBSDE) with jumps, focusing on cases where the mean reflection is nonlinear. Unlike traditional RBSDEs, this particular type of RBSDE imposes a constraint defined by the mean of a loss function that does not follow the continuous condition. We start by deriving an a priori estimate of the solution, followed by establishing the uniqueness and existence of the solution. Theoretical results are illustrated by way of an example of the application of super‐hedging to the reinsurance and investment problem under a risk constraint. [ABSTRACT FROM AUTHOR]

The main novelty of this paper lies in five aspects. (1) Our study on the fractional-order derivative stochastic Ginzburg–Landau equation (FODSGLE) has resulted in numerous exact solutions using Jacobian elliptic functions (JEFs). These solutions offer valuable insights into complex physical phenomena and have practical implications across various fields. (2) By examining the effect of noise on FODSGLE solutions, we found consistent behavior regardless of the form of fractional derivatives. As noise intensity increases, the solutions deteriorate and tend toward zero. This study contributes to the existing literature by providing new insights and surpassing previous efforts in understanding the dynamics of FODSGLE. (3) Our research investigates how fractional order affects noise in solutions of the FODSGLE. Surprisingly, we found that changes in the fractional order (α) have minimal influence on the system when the noise intensity (σ) is fixed. This indicates an aspect that has not been addressed extensively in existing literatures. (4) We compare different types of fractional derivatives within the context of the FODSGLE, presenting a novel contribution, while keeping the noise intensity or fractional order fixed. Our results demonstrate that the CFOD closely aligns with the MFOD, while displaying more differences with the solutions obtained using the BFOD. This comparison sheds light on an aspect that has not been thoroughly investigated in previous literatures. (5) This paper delves into the phase portraits of the FODSGLE and investigates the associated sensitivity and chaotic behaviors. Notably, previous studies on the stochastic Ginzburg–Landau equation (SGLE) have not extensively examined this aspect. By analyzing the phase portraits, we gain valuable insights into the dynamics and stability of FODSGLE solutions, uncovering intricate behaviors within the system. Our exploration of sensitivity and chaotic behaviors adds another dimension to understanding the (FODSGLE), laying a foundation for further research in this domain. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new method based on a library of chemical reactions for constructing a system of ordinary differential equations from stochastic simulations arising from an agent-based model. The advantage of this approach is that this library respects any coupling between systems components, whereas the SINDy algorithm (introduced by Brunton et al.) treats the individual components as decoupled from one another. Another advantage of our approach is that we can use a non-negative least squares algorithm to find the non-negative rate constants in a very robust, stable and simple manner. We illustrate our ideas on an agent-based model of tumour growth on a 2D lattice. [ABSTRACT FROM AUTHOR]

Autoregressive Markov switching (ARMS) time series models are used to represent real-world signals whose dynamics may change over time. They have found application in many areas of the natural and social sciences, as well as in engineering. In general, inference in this kind of systems involves two problems: (a) detecting the number of distinct dynamical models that the signal may adopt and (b) estimating any unknown parameters in these models. In this paper, we introduce a new class of nonlinear ARMS time series models with delays that includes, among others, many systems resulting from the discretisation of stochastic delay differential equations (DDEs). Remarkably, this class includes cases in which the discretisation time grid is not necessarily aligned with the delays of the DDE, resulting in discrete-time ARMS models with real (non-integer) delays. The incorporation of real, possibly long, delays is a key departure compared to typical ARMS models in the literature. We describe methods for the maximum likelihood detection of the number of dynamical modes and the estimation of unknown parameters (including the possibly non-integer delays) and illustrate their application with a nonlinear ARMS model of El Niño–southern oscillation (ENSO) phenomenon. [ABSTRACT FROM AUTHOR]

The realistic modeling of vibratory energy harvesters is necessary to ensure their reliable performance when subjected to random excitation. However, the hysteretic behavior of piezoelectric materials and the nonlocality of fractional electrical components make the numerical estimation of their stochastic response computationally challenging. This paper proposes an iterative method based on equivalent linearization and stochastic averaging for estimating the stationary probability density function of the output voltage amplitude of vibratory energy harvesters modeled as a hysteretic oscillator with fractional electrical components subject to a broadband stochastic process. By estimating the average of hysteretic terms, one can find an amplitude-dependent linear governing equation for the output voltage, from which one can determine a stochastic differential equation governing the evolution of the output voltage amplitude. We obtain the stationary solution of the corresponding Fokker–Planck equation in a semianalytical form, and we include comparisons with pertinent Monte Carlo simulation data to show that the estimated probability density functions are accurate and obtained at a minimal computational cost. [ABSTRACT FROM AUTHOR]

This paper studies stochastic control problems with the action space taken to be probability measures, with the objective penalized by the relative entropy. We identify a suitable metric space on which we construct a gradient flow for the measure-valued control process, in the set of admissible controls, along which the cost functional is guaranteed to decrease. It is shown that any invariant measure of this gradient flow satisfies the Pontryagin optimality principle. If the problem we work with is sufficiently convex, the gradient flow converges exponentially fast. Furthermore, the optimal measure-valued control process admits a Bayesian interpretation, which means that one can incorporate prior knowledge when solving such stochastic control problems. This work is motivated by a desire to extend the theoretical underpinning for the convergence of stochastic gradient type algorithms widely employed in the reinforcement learning community to solve control problems. [ABSTRACT FROM AUTHOR]

In [SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 2068--2092], Widiasih proposed and analyzed a deterministic one-dimensional Budyko--Sellers energy-balance model with a moving ice line. In this paper, we extend this model to the stochastic setting and analyze it within the framework of stochastic slow-fast systems. We derive the dynamics for the ice line in the limit of a small parameter as a solution to a stochastic differential equation. The stochastic approach enables the study of co-existing (metastable) climate states as well as the transition dynamics between them. [ABSTRACT FROM AUTHOR]

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The properties of a jump process is used. The stabilization of differential equations by stochastic feedback control is based on discrete-time state observations. Firstly, the stability results of the auxiliary system is established. Secondly, by comparing it with the auxiliary system and using the continuity method, the stabilization of the original system is obtained. Both parts focus on the impact of probability theory. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a feedback controller and an adaptive controller. Then, we obtain the synchronization criteria in a fixed time by combining the Lyapunov method and the related inequality theory of the stochastic differential equation and calculate the stabilization time for the STSFRNNS. In addition, to verify the authenticity of the theoretical results, we use MATLABR2023A to carry out numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we study large deviation principles of nonlinear filtering for McKean-Vlasov stochastic differential equations. First of all, we establish the large deviation principle for the space-distribution dependent Zakai equation by a weak convergence approach. Then based on the obtained result and the relationship between the space-distribution dependent Zakai equation and the space-distribution dependent Kushner-Stratonovich equation, the large deviation principle for the latter is proved. [ABSTRACT FROM AUTHOR]

In this paper, we consider the well-posedness for the anticipated backward stochastic Schrödinger equation in a bounded domain or the whole space $ \mathbb {R}^d $ R d , which is associated to a stochastic control problem of nonlinear Schrödinger equations with time delay effect. The approach to establish the existence and uniqueness of adapted solutions are mainly based on the complex Itô's formula, the Galerkin's approximation method and the martingale representation theorem. [ABSTRACT FROM AUTHOR]

This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy. [ABSTRACT FROM AUTHOR]

This paper proposed a Legendre-based multi-element probabilistic collocation method for time-dependent stochastic differential equations, used for density estimation of unknown functions. This method involves discretizing the stochastic space, and on each element, constructing Lagrange interpolation basis functions based on Legendre–Gauss–Lobatto collocation/quadrature nodes. The proposed method is applied to approximate one-dimensional/two-dimensional smooth/non-smooth functions and is tested for accuracy in approximating random function values, density estimations, and mathematical expectations. This method is applied to stochastic nonlinear Schrödinger equations and coupled stochastic nonlinear Schrödinger equations, and all numerical results are compared with Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

The operation of open source software (OSS) has a complex maintenance phase. This paper focuses on several irregular fluctuation in the operation performance of OSS. We apply a flexible jump diffusion process model to several noisy cases in the operation of OSS associated with version upgrade. In particular, we discuss a method of maintenance effort management based on the flexible jump diffusion process model considering the irregular fluctuation in version upgrade performance for OSS projects. Moreover, this paper proposes the optimal maintenance problems based on our flexible jump diffusion process models. Then, it will be useful for the OSS project managers to understand the effort status and the optimal maintenance times with version upgrade of OSS from the standpoint of OSS project management. Moreover, we analyze actual data to show numerical examples of our model and the proposed optimal maintenance problems considering the characteristics with version upgrade of OSS projects. [ABSTRACT FROM AUTHOR]

In this paper, we study the Wong-Zakai approximations of a stochastic lattice differential equation with long-range interactions and multiplicative white noise at each node. We first prove the existence and uniqueness of pullback random attractors for lattice system driven by multiplicative white noises as well as the corresponding Wong-Zakai approximate system. Then, we prove the convergence of solutions and the upper semicontinuity of random attractors for theWong-Zakai approximate system as the size of approximation approaches zero. [ABSTRACT FROM AUTHOR]

A balanced ecosystem with coexisting constituent species is often perturbed by different natural events that persist only for a finite duration of time. What becomes important is whether, in the aftermath, the ecosystem recovers its balance or not. Here we study the fate of an ecosystem by monitoring the dynamics of a particular species that encounters a sudden increase in death rate. For exploration of the fate of the species, we use Monte-Carlo simulation on a three-species cyclic rock-paper-scissor model. The density of the affected (by perturbation) species is found to drop exponentially immediately after the pulse is applied. In spite of showing this exponential decay as a short-time behavior, there exists a region in parameter space where this species surprisingly remains as a single survivor, wiping out the other two which had not been directly affected by the perturbation. Numerical simulations using stochastic differential equations of the species give consistency to our results. [ABSTRACT FROM AUTHOR]

Distribution-dependent stochastic differential equations (DDSDEs, in short), also called as McKean–Vlasov SDEs, are a special class of SDEs and have some concrete applications in mean-field control, mean-field games and complex networks. Stability plays fundamental role among the theory and applications for DDSDEs. In this paper, we discuss a class of distribution-dependent stochastic differential equations driven by time-changed Brownian motions (TDDSDEs, in short). The main aim of this paper is to establish the stability for TDDSDEs. Sufficient conditions are provided to guarantee the solutions to be stable in different senses with regard to a part of the variables in terms of a Lyapunov function. Two examples are given to show the usefulness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

To model dynamic systems in various situations results in an ordinary differential equation of the form d y d t = g (y , t , θ) , where g denotes a function and θ stands for a parameter or vector of unknown parameters that require estimation from observations. In order to consider environmental fluctuations and numerous uncontrollable factors, such as those found in forestry, a stochastic noise process ϵ t may be added to the aforementioned equation. Thus, a stochastic differential equation is obtained: d Y t d t = f (Y t , t , θ) + ϵ t . This paper introduces a method and procedure for parameter estimation in a stochastic differential equation utilising the Richards model, facilitating growth prediction in a forest's tree population. The fundamental concept of the approach involves assuming that a deterministic differential equation controls the development of a forest stand, and that randomness comes into play at the moment of observation. The technique is utilised in conjunction with the logistic model to examine the progression of an agricultural epidemic induced by a virus. As an alternative estimation method, we present the Random Time Transformation (RTT) method. Thus, this paper's primary contribution is the application of the RTT method to estimate the Richards model, which has not been conducted previously. The literature often uses the logistic or Gompertz models due to difficulties in estimating the parameter form of the Richards model. Lastly, we assess the effectiveness of the RTT Method applied to the Chapman–Richards model using both simulated and real-life data. [ABSTRACT FROM AUTHOR]

In this paper, we prove the hypoellipticity and parabolic hypoellipticity of nonlocal operators under Hörmander's condition. One key step in the proof is to obtain an off-diagonal small time estimate of the densities of the processes associated to these operators. To this end, we use a new version of Malliavin calculus on the Poisson space, namely the lent particle method, recently developed by Bouleau and Denis. [ABSTRACT FROM AUTHOR]

The mean square synchronization problem of the complex dynamical network (CDN) with the stochastic link dynamics is investigated. In contrast to previous literature, the CDN considered in this paper can be viewed as consisting of two subsystems coupled to each other. One subsystem consists of all nodes, referred to as the nodes subsystem, and the other consists of all links, referred to as the network topology subsystem, where the weighted values can quantitatively reflect changes in the network's topology. Based on the above understanding of CDN, two vector stochastic differential equations with Brownian motion are used to model the dynamic behaviors of nodes and links, respectively. The control strategy incorporates not only the controller in the nodes but also the coupling term in the links, through which the CDN is synchronized in the mean-square sense. Meanwhile, the dynamic stochastic signal is proposed in this paper, which is regarded as the auxiliary reference tracking target of links, such that the links can track the reference target asymptotically when synchronization occurs in nodes. This implies that the eventual topological structure of CDN is stochastic. Finally, a comparison simulation example confirms the superiority of the control strategy in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with a class of delay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 ). In this type of equation, a generator at time t can depend not only on the present but also on the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is a divergence-type integral. [ABSTRACT FROM AUTHOR]

This paper studies an insurer's optimal investment portfolio under the mean-variance criterion. The financial market consists of a riskless bond and a risky asset, and the latter's volatility is random. We are extending the Cox–Ingersoll–Ross (CIR) model to the case with jumps, where it is modeled by a jump-diffusion stochastic differential equation (SDE). We use a Lévy SDE to describe the risk process we have, in which we extend the classic Cramér-Lundberg model to the Lévy process, and additionally introduce the stochastic volatility into this model. We assume that the insurer in question is a mean-variance optimizer. In other words, the decision that this insurer faces is to simultaneously maximize and minimize the mean and variance of his/her terminal wealth by selecting an optimal portfolio. We have uncovered closed-form solutions to the mean-variance problem with respect to the efficient strategy and efficient frontier by solving for expected utility maximization of a quadratic function through the martingale method. Finally, we give a numerical example that analyzes the economic behavior of the efficient frontier. [ABSTRACT FROM AUTHOR]

In this paper we propose a simple dynamic model of tumor-immune system. Such a model is able to describe the dynamics of tumor cell population under the influence of random fluctuations and is in the form of a stochastic differential equation driven by multiplicative Gaussian noise. We analyze the model and establish necessary conditions for extinction and persistence of tumor cells. The results are illustrated by computer simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and exponential stability in pth moment of mild solutions for a class of impulsive fractional stochastic differential equations driven by Poisson jumps. Firstly, we discuss the existence and uniqueness of mild solutions for the considered equations by the Banach fixed point theorem. Next, we establish a new impulsive-integral inequality that can effectively improve some previous results [4, 17, 5, 3, 6]. Then, we obtain the exponential stability in the pth moment of mild solutions for the considered equations with the aid of the new inequality. Finally, an example is given to illustrate the efficiency of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Stochastic Differential Equations (SDEs) describe physical systems to account for random forcing terms in the evolution of the state trajectory. The noisy sampling mixer, a component of digital wireless communications, can be regarded as a potential case from the dynamical systems' viewpoint. The universality of the noisy sampling mixer is attributed to the fact that it adopts the structure of a nonlinear SDE and its linearized version becomes a time-varying bilinear SDE. This paper develops a mathematical theory for the nonlinear noisy sampling mixer from the filtering viewpoint. Since the filtering of stochastic systems hinges on the structure of dynamical systems and observation equation set up, we consider three 'filtering models'. The first model, accounts for a nonlinear SDE coupled with a nonlinear observation equation. In the second model, we consider a bilinear SDE with a linear observation equation to achieve the nonlinear sampling filtering. Note that the bilinear SDE coupled with the linear observation is a consequence of the Carleman linearization to the nonlinear SDE and the nonlinear observation equation. In the third model, we consider a Stratonovich SDE coupled with a nonlinear observation equation. The filtering equation of this paper can be further utilized to guide the design process of the noisy sampling mixer. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence and uniqueness of a solution for a stochastic differential equation driven by an additive noise which is the sum of two fractional Brownian motions with different Hurst parameters. The proofs are based on the techniques of fractional calculus and Girsanov theorem. In particular, we show that the regularization effect of the fractional Brownian motion with the smaller Hurst index dominates. A key challenge in this paper is to extend and apply the Girsanov theorem for two noises given by the sum of two (dependent) fractional Brownian motions by using profound techniques of fractional operator theory. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the behavior of a linear classical parametric amplifier in the presence of white noise and give theoretical estimates of the noise spectral density based on approximate Green's functions obtained by using averaging techniques. To validate our theory, we compare the analytical results with experimental data from an analog circuit and with numerical simulations of the model's stochastic differential equations. The experimental data were accurately described by our model. Moreover, we noticed spectral components in the output signal of the amplifier, which are due to noisy precursors of instability. The position, width, and magnitude of these components are in agreement with the noise spectral density obtained by the theory proposed here. [ABSTRACT FROM AUTHOR]

This paper is concerned with quantum stochastic differential equations driven by the fermion field in noncommutative space Lp(풞) for 2≤p<∞$$ 2\le p<\infty $$. First, we investigate the existence and uniqueness of Lp$$ {L}&#x0005E;p $$‐solutions of quantum stochastic differential equations in an infinite time horizon by using the noncommutative Burkholder–Gundy inequality given by Pisier and Xu and the noncommutative generalized Minkowski inequality. Then, we investigate the stability, self‐adjointness, and Markov properties of Lp$$ {L}&#x0005E;p $$‐solutions and analyze the error of numerical schemes of quantum stochastic differential equations. The results of this paper can be utilized for investigating the optimal control of quantum stochastic systems with infinite dimensions. [ABSTRACT FROM AUTHOR]

The classical Lotka–Volterra predator–prey model is globally stable and uniformly persistent. However, in real-life biosystems, the extinction of species due to stochastic effects is possible and may occur if the magnitudes of the stochastic effects are large enough. In this paper, we consider the classical Lotka–Volterra predator–prey model under stochastic perturbations. For this model, using an analytical technique based on the direct Lyapunov method and a development of the ideas of R.Z. Khasminskii, we find the precise sufficient conditions for the stochastic extinction of one and both species and, thus, the precise necessary conditions for the stochastic system's persistence. The stochastic extinction occurs via a process known as the stabilization by noise of the Khasminskii type. Therefore, in order to establish the sufficient conditions for extinction, we found the conditions for this stabilization. The analytical results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper considers a pair (F , τ) , where F is a filtration representing the "public" flow of information that is available to all agents over time, and τ is a random time that might not be an F -stopping time. This setting covers the case of a credit risk framework, where τ models the default time of a firm or client, and the setting of life insurance, where τ is the death time of an agent. It is clear that random times cannot be observed before their occurrence. Thus, the larger filtration, G , which incorporates F and makes τ observable, results from the progressive enlargement of F with τ. For this informational setting, governed by G , we analyze the optimal stopping problem in three main directions. The first direction consists of characterizing the existence of the solution to this problem in terms of F -observable processes. The second direction lies in deriving the mathematical structures of the value process of this control problem, while the third direction singles out the associated optimal stopping problem under F. These three aspects allow us to deeply quantify how τ impacts the optimal stopping problem and are also vital for studying reflected backward stochastic differential equations that arise naturally from pricing and hedging of vulnerable claims. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE. [ABSTRACT FROM AUTHOR]

First hitting times (FHTs) describe the time it takes a random "searcher" to find a "target" and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order s ε (0, 1) (describing a (2s)-stable Lévy flight whose squared displacement scales as t1/s in time t) and targets of radius \varepsilon \ll 1, we show that the MFHT is order one for s ε (1/2, 1) and diverges as log(1/\varepsilon) for s = 1/2 and \varepsilon 2s 1 for s ε (0, 1/2). We then use our asymptotic results to identify the value of s ε (0, 1] which minimizes the average MFHT and find that (a) this optimal value of s vanishes for sparse targets and (b) the value s = 1/2 (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations. [ABSTRACT FROM AUTHOR]

There are numerous phenomena in real world that their practical modeling in mathematics language deals with differential equations. Stochastic terms have significant roles in estimating behavior of these events such as machine tool vibrations, biology, traffic dynamics, neural networks and so on, it cannot be reasonable to consider an event without its past information, and it ends up with equations including time delays. The numerical solution of stochastic delay differential equations has always been of interest to researchers. In this paper, by using scaling function of the Daubechies wavelet, a collocation method is presented to solve a class of stochastic delay differential equations and it is tried to demonstrate the advantages of using this type of wavelet. Convergence of the method is provided and numerical experiments are reported to show application of the method in practice. [ABSTRACT FROM AUTHOR]

In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Hölder inequality and Grönwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories. [ABSTRACT FROM AUTHOR]

The transition kernel of an ℝn-valued diffusion or jump diffusion process {Xt} is known to satisfy the Feller property if {Xt} is the solution of an SDE whose coefficients are Lipschitz continuous. This Lipschitz route to Feller falls short if {Xt} is the solution of an SDE whose coefficients depend on a state-dependent regime-switching process {θt}. In this paper it is shown that pathwise uniqueness and the Feller property are satisfied under mild conditions for a regime-switching jump diffusion process {Xt, θt} with hybrid jumps, i.e. jumps in {Xt} that occur simultaneously with {θt} switching. [ABSTRACT FROM AUTHOR]

A mathematical justification of some basic structural properties of stochastically perturbed gene regulatory networks, including those with autoregulation and delay, is offered in this paper. By using the theory of stochastic differential equations, it is, in particular, shown how to control the asymptotic behavior of the diffusion terms in order to not destroy certain qualitative features of the networks, for instance, their sliding modes. The results also confirm that the level of randomness is gradually reduced if the gene activation times become much smaller than the time of interaction of genes. Finally, the suggested analysis explains why the deterministic numerical schemes based on replacing smooth, steep response functions by the simpler yet discontinuous Heaviside function, the well-known simplification algorithm, are robust with respect to uncertainties in data. The main technical difficulties of the analysis are handled by applying the uniform version of the stochastic Tikhonov theorem in singular perturbation analysis suggested by Yu. Kabanov and S. Pergamentshchikov. [ABSTRACT FROM AUTHOR]