Theory Of Stochastic Differential Equations Research Articles

Consensus for discrete-time second-order multi-agent systems in the presence of noises and semi-Markovian switching topologies

In this paper, we focus on the control problem of mean square consensus for second-order continuous-time multi-agent systems with multiplicative noises under Markovian switching topologies. A new Lyapunov function based on the Laplacian matrix of the corresponding union topology of all possible topologies is designed for the stochastic stability analysis of consensus. Applying matrix theory and stochastic stability for stochastic differential equations, we can analyze the consensus problem of the considered systems with the designed Lyapunov function. In addition, we present the sufficient conditions of the mean square consensus exponentially for the considered stochastic systems. Finally, we give an simulation example to numerically validate our theoretical results.

State estimation based on measurement from a part of nodes for a delayed Itô type of complex network with Markovian mode-dependent parameters

This article is devoted to coping with the state estimate problem for a class of Itô stochastic complex network. The target network under consideration is a hybrid system involving Markovian jumping parameters and mode-dependent distributed time-delays, and the system noise is characterized by the Brownian Motion. The problem addressed is to design a state estimator to track the states of target network under the assumption that the network outputs are from only a portion of network nodes. By utilizing the Lyapunov stability method and the Itô stochastic differential equation theory, sufficient conditions are educed so that the state estimation error of the estimator is mean-square exponentially ultimately bounded. In particular, in the case that the system is noise-free, the derived conditions can ensure the state estimation error is mean-square exponentially stable. Furthermore, the estimator design can be made by solving a set of matrix inequalities. Lastly, a numerical example is exploited to indicates the validity of the theoretical results.

The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets

The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket.

Fluctuation theorem as a special case of Girsanov theorem

Stochastic thermodynamics is an important development in the direction of finding general thermodynamic principles for non-equilibrium systems. We believe stochastic thermodynamics has the potential to benefit from the measure-theoretic framework of stochastic differential equations (SDEs). Toward this, in this work, we show that fluctuation theorem (FT) is a special case of the Girsanov theorem, which is an important result in the theory of SDEs. We report that by employing Girsanov transformation of measures between the forward and the reversed dynamics of a general class of Langevin dynamic systems, we arrive at the integral fluctuation relation. Following the same approach, we derive the FT also for the overdamped case. Our derivation is applicable to both transient and steady state conditions and can also incorporate diffusion coefficients varying as a function of state and time, e.g. in the context of multiplicative noise. We expect that the proposed method will be an easy route towards deriving the FT irrespective of the complexity and non-linearity of the system.

Anticipating regime shifts by mixing early warning signals from different nodes

Real systems showing regime shifts, such as ecosystems, are often composed of many dynamical elements interacting on a network. Various early warning signals have been proposed for anticipating regime shifts from observed data. However, it is unclear how one should combine early warning signals from different nodes for better performance. Based on theory of stochastic differential equations, we propose a method to optimize the node set from which to construct an early warning signal. The proposed method takes into account that uncertainty as well as the magnitude of the signal affects its predictive performance, that a large magnitude or small uncertainty of the signal in one situation does not imply the signal’s high performance, and that combining early warning signals from different nodes is often but not always beneficial. The method performs well particularly when different nodes are subjected to different amounts of dynamical noise and stress.

Bogoliubov-type recursions for renormalisation in regularity structures

Hairer's regularity structures transformed the theory of singular stochastic partial differential equations and their solutions. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation procedures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. This work revisits the latter by defining Bogoliubov-type recursions similar to Connes and Kreimer's formulation of BPHZ renormalisation. We then apply our approach to the renormalisation problem for SPDEs.

Developing a concept for the economic and social management of a city with due account for the predicted industrial development

Subject. Theoretical, methodological, and practical issues related to the economic and social management of the city with due account of the predicted values of the industrial development.Objectives. Justification of scientific and methodological provisions of a conceptual nature related to urban management and solving local social problems by forecasting the level of industrial development.Methodology. The authors put forward a hypothesis: if the level of industrial development is forecasted accurately, it is possible to plan upcoming changes in the main economic and social indicators of the municipal economy. Theoretically, the hypothesis is based on the method of logical scaling and synthesis of scientific and practical knowledge. Statistical studies were conducted using the method of econometric and financial analysis based on socio-economic indicators of industrial cities in the Southern Federal District of Russia for the period from 2013 to 2022. Forecasting for the subsequent periods is based on an applied approach to constructing estimates of the quasi-maximal likelihood of the level of industrial development, which was carried out using the theory of stochastic differential equations based on methods of probabilistic forecasting.Results and discussion. As a result of the study, it was substantiated that municipal management contributes to the production of material goods and to harmonisation of socio-economic relations between local authorities, businesses, public organisations, and individuals, which are interrelated with the level of industrial development of a city. The developed applied approach to constructing estimates of the quasi-maximal likelihood of values of socio-economic parameters makes it possible to forecast the level of industrial development of cities in the Russian Federation. Point and interval forecasts lay the foundation for planning economic and other indicators. The concept of economic and social management of administrative-territorial units was proposed. The mandatory socio-economic blocks of the concept were supplemented with the functionality of prediction modelling, which requires further discussion, critical understanding, and subsequent improvement. Future changes in local industrial policy need to be analysed and evaluated in terms of their effectiveness in the context of structural modernisation of the economic and organisational mechanism for urban and social management of the city.

Mean-variance reinsurance and asset liability management with common shock via non-Markovian stochastic factors

This paper explores a reinsurance and asset liability management problem in a factor model with dependent risks, considering economic factors as non-Markovian processes impacting the appreciation rate, volatility, and value of liabilities. It encompasses various stochastic volatility (SV) models like Hull-White, Heston, 3/2, and 4/2 models as special cases. The model addresses dependent risks through a common shock in claim number processes. The insurer's aim is to minimize variance in terminal net wealth while achieving a certain expected terminal net wealth, balancing reinsurance, new business, and asset allocation. The paper employs linear-quadratic (LQ) control and backward stochastic differential equation (BSDE) theory to derive both the efficient strategy and the efficient frontier. To illustrate our results, numerical examples are provided in two special cases, the 3/2 and 4/2 SV models.

Effects of noise on the critical points of Turing instability in complex ecosystems.

Noise is ubiquitous in natural and artificial systems. In a noisy environment, the interactions among nodes may fluctuate randomly, leading to more complicated interactions. In this paper we focus on the effects of noise and network topology on the Turing pattern of ecological networks with activator-inhibitor structure, which may be interpreted as prey-predator interactions. Based on the stability theory of stochastic differential equations, a sufficient condition for the uniform state is derived. The analytical results indicate that noise is beneficial for the uniform state. When the ratio between the diffusion coefficients of the predator and prey increases, the ecosystems can exhibit a transition from a uniform stable state to a Turing pattern, while when the ratio decreases, the ecosystems transit from a Turing pattern to a uniform stable state. There are two crucial critical points in Turing patterns, forward and backward. We find that both forward and backward critical points increase as the noise intensity increases. This means that noise favors a stable homogeneous state compared to a state with a heterogeneous pattern, which is consistent with the analytical results. In addition, noise can weaken the hysteresis phenomenon and even eliminate it in some cases. Furthermore, we report that network topology plays an important role in modulating the uniform state of ecosystems, such as the size of prey-predator systems, the network connectivity, and the strength of interaction.

Global dynamics of a stochastic reaction–diffusion predator–prey system with space-time white noise

This paper proposes a stochastic reaction–diffusion predator–prey system with fear under the interference of space-time white noise by using the related theories of stochastic partial differential equations. The motivation of this paper is twofold: (i) mathematically, to try to investigate the effects of environmental noise and diffusion on population dynamics; (ii) biologically, to study how environmental noise affects the permanence of species. First, we analyse the well-posedness of solutions. Then sufficient conditions for extinction and permanence of prey and predator populations are derived. Moreover, we present the existence and uniqueness of stationary distribution of this system. Finally, based on numerical simulations and theoretical analysis, it is revealed that (i) high-intensity white noise can lead to the extinction of predator populations; (ii) low-intensity white noise may prolong the period of periodic solutions. This study provides new theoretical guidance for exploring ecological issues in the face of environmental disturbance and spatial diffusion.

General path integrals and stable SDEs

The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to \operatorname{d} Z_t=\sigma(Z_{t-})\operatorname{d} X_t driven by a two-sided \alpha -stable Lévy process, in the spirit of the classical Engelbert–Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence and uniqueness of weak solutions for \alpha\in(0,1) . Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We prove integral tests for finiteness of path integrals under minimal assumptions.

A Sobolev Space Theory for Time-Fractional Stochastic Partial Differential Equations Driven by Lévy Processes

A Sobolev Space Theory for Time-Fractional Stochastic Partial Differential Equations Driven by Lévy Processes

Distributed Observers for State Omniscience with Stochastic Communication Noises

The focus of this paper is on solving the state estimation problem for general continuous-time linear systems through the use of distributed networked observers. To better reflect the communication environment, stochastic noises are considered when observers exchange information. In the networked observers, each local observer measures only part of the system output, and the state estimation can not be accomplished within a single observer. Then, all observers communicate through a pre-specified graph to make up information in the remaining system output. By solving a parametric algebraic Riccati equation (ARE), a simple method to calculate parameters in the observers is proposed. Furthermore, using the stability theory of stochastic differential equations, state omniscience is discussed in almost sure sense and in the mean square sense for the cases of state-dependent noises and non-state-dependent noises, respectively. It is shown that, for observable linear systems, the resulting observers work in a coordinated mode to reach state omniscience under the connected graph. Illustrative examples are provided to show the effectiveness of the distributed observers.

Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

We derive the fluctuation-dissipation relation and explore its connection with the equipartition theorem and Maxwell-Boltzmann statistics through the use of different stochastic analytical techniques. Our first approach is the theory of backward stochastic differential equations, which arises naturally in this context, and facilitates the understanding of the interplay between these classical results of statistical mechanics. Moreover, it allows to generalize the classical form of the fluctuation-dissipation relation. The second approach consists in deriving forward stochastic differential equations for the energy of an electric system according to both Itô and Stratonovich stochastic calculus rules. While the Itô equation possesses a unique solution, which is the physically relevant one, the Stratonovich equation admits this solution along with infinitely many more, none of which has a physical nature. Despite of this fact, some, but not all of them, obey the fluctuation-dissipation relation and the equipartition of energy.

Modeling the Effects of Chemotherapeutic Dose Response on a Stochastic Tumor-Immune Model of Prostate Cancer with Androgen Deprivation Therapy

The periodical application of androgen deprivation therapy, immunotherapy, or chemotherapy is an effective method for cancer treatment, but few studies combine them. To explore such comprehensive treatment mechanisms, this paper establishes a pulsed stochastic hybrid dynamics model considering tumor antigenicity and density-dependent mortality. In addition to analyzing the basic properties of solutions such as the tumor-free periodic solution and global attraction of the model, the threshold conditions for the persistence and extinction of prostate cancer cells and effector cells are obtained by using stochastic differential equation theory. Besides, sufficient conditions for the existence of stationary distribution of the system are established. The results reveal that comprehensive therapy or white noise can determine tumor dynamics and suggest that the treatment of prostate cancer should be individualized according to the state of tumor development. Finally, biological significance is discussed and conclusions are given.

Three-Party Stochastic Evolutionary Game Analysis of Supply Chain Finance Based on Blockchain Technology

In the process of accounts receivable financing under supply chain finance, the phenomenon of accounts receivable forgery and default have caused great pressure on the supervision of financial institutions. We consider the integration of blockchain technology with a supply chain finance platform around the fraudulent default phenomenon in supply chain finance receivables financing and construct a three-party stochastic evolutionary game model among financial institutions, core enterprises, and Micro, Small, and Medium Enterprises (MSMEs). Firstly, we use Ito^’s stochastic differential equation theory to analyze the conditions for the stability of the behavior of game subjects. Secondly, we use numerical simulations to quantitatively analyze the impact of the regulatory strength of financial institutions, the information sharing of the blockchain platform, and the change of incentive parameters on the strategy choice of game subjects. Through the above analysis, we conclude that the information-sharing incentive coefficient promotes financial institutions to choose to connect to the blockchain platform, and the information-sharing risk coefficient and the regulatory intensity have the opposite effect on the blockchain platform construction. Meanwhile, the allocation of incentive shares has a significant influence on the core enterprises. Finally, we give priorities and directions for adjusting the relevant parameters to provide recommendations for financial institutions to regulate the financing process more effectively.

Stratifications and foliations in phase portraits of gene network models.

Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section "Investigation of functions and plot of their graphs" of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations.

The evolution to equilibrium of solutions to nonlinear Fokker-Planck equation

One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$ u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, t\geq0, x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and $b$ the convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, for some $t_n\to\infty$ of the solution $u(t_n)$ to an equilibrium state of the equation for a large set of nonnegative initial data in $L^1$. These results are new in the literature on nonlinear Fokker-Planck equations arising in the mean field theory and are also relevant to the theory of stochastic differential equations. As a matter of fact, by the above convergence result, it follows that the solution to the McKean-Vlasov stochastic differential equation corresponding to (1), which is a nonlinear distorted Brownian motion, has this equilibrium state as its unique invariant measure. Keywords: Fokker-Planck equation, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, McKean-Vlasov stochastic differential equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject Classification: 35B40, 35Q84, 60H10.

Time Cost for Consensus of Stochastic Multiagent Systems With Pinning Control

In this article, the estimation of time cost for stochastic consensus of second-order nonlinear multiagent systems (MASs) is investigated. Compared with previous studies, the effect of noise and nonlinear inherent dynamics of agents are considered. In order to achieve the finite/fixed-time stochastic consensus while minimizing energy consumption, the pinning protocols are designed, which only need to control a small fraction of agents. Sufficient conditions for the stochastic consensus are established by employing the stability theory of stochastic differential equations and the algebra graph theory. Finally, the proposed protocols are verified through numerical simulation on the consensus of unmanned aerial vehicles (UAVs) system.

Event-Triggered Finite-Time <i>H</i><sub>∞</sub> Filtering for Discrete-Time Nonlinear Stochastic Systems

This paper addresses the problem of event-triggered finite-time H∞ filter design for a class of discrete-time nonlinear stochastic systems with exogenous disturbances. The stochastic Lyapunov-Krasoviskii functional method is adopted to design a filter such that the filtering error system is stochastic finite-time stable (SFTS) and preserves a prescribed performance level according to the pre-defined event-triggered criteria. Based on stochastic differential equations theory, some sufficient conditions for the existence of H∞ filter are obtained for the suggested system by employing linear matrix inequality technique. Finally, the desired H∞ filter gain matrices can be expressed in an explicit form.