System Of Stochastic Differential Equations Research Articles

Moment Lyapunov exponents and stochastic nonstability of a circular cylindrical shells

The paper investigates the stochastic stability conditions of circular cylindrical shells, compressed by time-dependent stochastic membrane forces. The main contribution of the paper is the determination of the moment Lyapunov exponents of the linear cylindrical shell for the first time. The moment and almost-sure stochastic stability of a three-degree-of-freedom coupled continuous system, under parametric excitation of white noise, are investigated. It is assumed that the system possesses a small damping. This type of a shell structure model requires a specific scheme for determining the moment Lyapunov exponents. The system of stochastic differential equations of motion is decoupled by using the contact transformation method with the obtained appropriate symplectic matrix of the system. Then a scheme for determining the moment Lyapunov exponents is presented which contains an analytical procedure. The largest Lyapunov exponent is calculated through its relation with the moment Lyapunov exponent. The moment and almost-sure stability boundaries are obtained analytically. These results are important and useful in engineering applications as an example of a procedure for dynamic analysis of mechanical systems with coupled stochastic oscillators.

Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network

We discuss the analysis of a piezoelectric energy harvester for random mechanical vibrations, and we assess the performance improvement guaranteed by interposing a matching network between the transducer and the electrical load, in terms of average output power and power efficiency. The mathematical model describing the harvester is a system of stochastic differential equations, where both cases of linear and nonlinear devices are considered. In the linear case, the power delivered to the load is increased by a factor of about 20 with respect to the direct connection, with a similar increase in the conversion efficiency. In the nonlinear case, we use a moment closure technique to calculate the first- and second-order moments of the electro-mechanical variables in the weak noise limit. Moment calculation is used to determine the optimal values of the matching network components that maximize the performance. In the strong noise limit, the state equations are integrated numerically to determine the same performance metrics. Our analysis shows that a properly designed matching network improves the performance by a significant amount, especially at low noise intensity.

System of Non-Linear Stochastic Differential Equations with Financial Market Quantities

In this paper, two systems of modified stochastic differential equations were considered. The variable coefficient problem was solved using Ito’s theorem to obtain an analytical solutions which was used to generate various behaviors of asset values which shows as follows: (i) increase in when are fixed increases the value of asset returns. (ii) a little increase on time when return rates and stock volatility are fixed increases the value of assets.(iii) an increase in the volatility parameter increases the value of asset pricing and parameter shows the various levels of long term investment plans, (iv) increase in rate of mean-reversion parameter reduces the value of asset. (v) An increase in the volatility parameter decreases the value of asset pricing (vi) The goodness of fit probability QQplots are not statistically significant and besides do come from a common distribution which has a vital meaning in the assessment of asset values for capital market investments. Nevertheless, the Tables 1,2 and 3 are best in comparisons with Tables 4,5 and 6 in terms of predictions for capital investments. The governing investment equations are unique and therefore are found to be satisfactory.

Neural parameter calibration for large-scale multiagent models

Computational models have become a powerful tool in the quantitative sciences to understand the behavior of complex systems that evolve in time. However, they often contain a potentially large number of free parameters whose values cannot be obtained from theory but need to be inferred from data. This is especially the case for models in the social sciences, economics, or computational epidemiology. Yet, many current parameter estimation methods are mathematically involved and computationally slow to run. In this paper, we present a computationally simple and fast method to retrieve accurate probability densities for model parameters using neural differential equations. We present a pipeline comprising multiagent models acting as forward solvers for systems of ordinary or stochastic differential equations and a neural network to then extract parameters from the data generated by the model. The two combined create a powerful tool that can quickly estimate densities on model parameters, even for very large systems. We demonstrate the method on synthetic time series data of the SIR model of the spread of infection and perform an in-depth analysis of the Harris-Wilson model of economic activity on a network, representing a nonconvex problem. For the latter, we apply our method both to synthetic data and to data of economic activity across Greater London. We find that our method calibrates the model orders of magnitude more accurately than a previous study of the same dataset using classical techniques, while running between 195 and 390 times faster.

Scaling limits of bisexual Galton–Watson processes

Bisexual Galton–Watson processes are discrete Markov chains where reproduction events are due to mating of males and females. Owing to this interaction, the standard branching property of Galton–Watson processes is lost. We prove tightness for conveniently rescaled bisexual Galton–Watson processes, based on recent techniques developed in [V. Bansaye, M.E. Caballero, and S. Méléard, Scaling limits of population and evolution processes in random environment, Electron. J. Probab. 24(19) (2019), pp. 1–38]. We also identify the possible limits of these rescaled processes as solutions of a stochastic system, coupling two equations through singular coefficients in Poisson terms added to square roots as coefficients of Brownian motions. Under some additional integrability assumptions, pathwise uniqueness of this limiting system of stochastic differential equations and convergence of the rescaled processes are obtained. Two examples corresponding to mutual fidelity are considered.

Unified Moment-Based Modeling of Integrated Stochastic Processes

Dial M for Simulation For years, systems of stochastic differential equations (SDEs) were simulated by discretization, inevitably introducing a bias, which can be difficult to quantify accurately. To circumvent this, some attempts have been made to simulate exactly various models from the SDE solution. These approaches prove capable of producing accurate results. A serious drawback of such an approach, nevertheless, is the implicit need to use extensive numerical methods, which make the entire simulation computationally heavy and quite impracticable. In the paper “Unified moment-based modeling of integrated stochastic processes,” Kyriakou, Brignone, and Fusai present a methodological framework based on M(oments) for the simulation of such models that overcomes earlier limitations. Theoretical results and extensive numerical experiments show that the proposed approach allows accurate simulation of complex stochastic models with low computational effort.

A novel mathematical analysis and threshold reinforcement of a stochastic dengue epidemic model with Lévy jumps

The aggressive re-emergence of dengue fever during the last decade made it a principal topic of several research fields, especially biomathematics. In this vein, mathematical modeling is commonly viewed as a great option to get an idea of this disease’s prevalence behavior. This work presents and analyzes a generalized stochastic dengue model that incorporates slight and huge environmental perturbations. In rigorous terms, our proposed model takes the form of a stochastic differential equations system that includes white noises and Lévy jumps. We first demonstrate its well-posedness then, and based on many novel analytical techniques, we provide some ameliorated necessary conditions for the extinction and the persistence in the mean. The theoretical findings show that the dynamics of our disturbed dengue model are mainly determined by the parameters related to the white noises and Lévy jumps intensities. In the end, certain numerical illustrative examples are depicted to confirm our findings and to highlight the effect of the adopted mathematical techniques on the results.

A Brownian–Markov stochastic model for cart-like wheeled mobile robots

Wheeled mobile robots are commonly used in a wide range of applications from automated warehouses to patrolling. A rigorous and accurate model describing their dynamics is then important for control and tracking. It is worth mentioning that the majority of these models in literature deal with deterministic settings. Therefore, they are not able to take into account stochastic uncertainty or non-predictable phenomena such as lateral grip loss. In this paper, we present a novel stochastic dynamic model which considers random perturbations, while also considering the effect of unknown dissipative external forces acting on wheeled mobile robots. In particular, our approach is based on a two-state hybrid system of Stochastic Differential Equations modeling the robot dynamics subject to Brownian motion noises, with transitions from one state to the other triggered by to a homogeneous Markov chain.

Stationary solutions and local equations for interacting diffusions on regular trees

We study the invariant measures of infinite systems of stochastic differential equations (SDEs) indexed by the vertices of a regular tree. These invariant measures correspond to Gibbs measures associated with certain continuous specifications, and we focus specifically on those measures which are homogeneous Markov random fields. We characterize the joint law at any two adjacent vertices in terms of a new two-dimensional SDE system, called the “local equation”, which exhibits an unusual dependence on a conditional law. Exploiting an alternative characterization in terms of an eigenfunction-type fixed point problem, we derive existence and uniqueness results for invariant measures of the local equation and infinite SDE system. This machinery is put to use in two examples. First, we give a detailed analysis of the surprisingly subtle case of linear coefficients, which yields a new way to derive the famous Kesten-McKay law for the spectral measure of the regular tree. Second, we construct solutions of tree-indexed SDE systems with nearest-neighbor repulsion effects, similar to Dyson’s Brownian motion.

The environmental effect on dynamics of the competition model with herd behavior

In this paper, the competition model with herd behavior and the Allee effect, which dynamics is described by the two-dimensional system of stochastic differential equations, is considered. We verify that the environmental noise included in the model provides a solution that is positive, global and bounded. Conditions under which the system is stable in some sense are presented. Finally, as an illustration, we apply our mathematical results for the prediction of time for which populations of the Asiatic black bear (Ursus thibetanus) and wild boar (Sus scrofa) in the Mount Jiri National Park, need to reach their equilibrium state.

Вимирання та виживання у залежній від щільності популяціії хижака стохастичній моделі хижак-жертва із стрибками

The non-autonomous stochastic density dependent predator-prey model with Holling-type II functional response disturbed by white noise, centered and non-centered Poisson noises is investigated. Corresponding system of stochastic differential equations has a unique, positive, global (no explosions in a finite time) solution. Sufficient conditions are obtained for extinction, non-persistence in the mean, weak and strong persistence in the mean of a predator and prey population densities in the considered stochastic predator-prey model.

Analysis of a stochastic epidemic model for cholera disease based on probability density function with standard incidence rate

<abstract><p>Acute diarrhea caused by consuming unclean water or food is known as the epidemic cholera. A model for the epidemic cholera is formulated by considering the instants at which a person contracts the disease and the instant at which the individual exhibits symptoms after consuming the poisoned food and water. Initially, the model is formulated from the deterministic point of view, and then it is converted to a system of stochastic differential equations. In addition to the biological interpretation of the stochastic model, we proved the existence of the possible equilibria of the associated deterministic model, and accordingly, stability theorems are presented. It is demonstrated that the proposed stochastic model has a unique global solution, and adequate criteria are constructed by using the Lyapunov function theory, which guarantees that the system has persistence in the mean whenever $ {\bf{R_s^0}} > 1 $. For the case of $ R_s < 1 $, we proved that the disease will tend to be eliminated from the community. Some graphical solutions were produced in order to better validate the analytical results that were acquired. This research can offer a solid theoretical foundation for comprehensive knowledge of other chronic communicable diseases. Additionally, our approach seeks to offer a technique for creating Lyapunov functions that may be utilized to investigate the stationary distributions of models with non-linear stochastic perturbations.</p></abstract>

Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model

<p style='text-indent:20px;'>This paper is devoted to study the open-loop equilibrium strategy for a mean-variance portfolio problem with investment constraints in a non-Markovian regime-switching jump-diffusion model. Specially, the investment strategies are constrained in a closed convex cone and all coefficients in the model are stochastic processes adapted to the filtration generated by a Markov chain. First, we provide a necessary and sufficient condition for an equilibrium strategy, which involves a system of forward and backward stochastic differential equations (FBSDEs, for short). Second, by solving these FBSDEs, we obtain a feedback representation of the equilibrium strategy. Third, we prove a theorem ensuring the almost everywhere uniqueness of the equilibrium solution. Finally, the results are applied to solve an example of the Markovian regime-switching model.</p>

Approximations for adapted M-solutions of type-II backward stochastic Volterra integral equations

In this paper, we study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, we obtain two approximation results, namely, a BSDE approximation and a numerical approximation. The BSDE approximation means that the solution of a finite system of backward stochastic differential equations (BSDEs) converges to the adapted M-solution of the original equation. As a consequence of the BSDE approximation, we obtain an estimate for the L2-time regularity of the adapted M-solutions of Type-II BSVIEs. For the numerical approximation, we provide a backward Euler-Maruyama scheme, and show that the scheme converges in the strong L2-sense with the convergence speed of order 1/2. These results hold true without any differentiability conditions for the coefficients.

A New Proof of the Expansion of Iterated Ito Stochastic Integrals with Respect to the Components of a Multidimensional Wiener Process Based on Generalized Multiple Fourier Series and Hermite Polynomials

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality, namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.

Stochastic linear-quadratic control with a jump and regime switching on a random horizon

In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a random horizon $ [0,T\wedge\tau] $, where $ \tau $ is a given random jump time for the underlying state process and $ T $ a constant. We obtain the explicit optimal feedback control and explicit optimal value of the problem by solving a system of stochastic Riccati equations (SREs) with jumps on the random horizon $ [0,T\wedge\tau] $. By the decomposition approach stemming from filtration enlargement theory, we express the solution to the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon $ [0,T] $. Solving the latter system is the key theoretical contribution of this paper and we accomplish this under three different conditions, one of which seems to be new in the literature. The above results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.

Global existence for quadratic FBSDE systems and application to stochastic differential games

In this note, we extend some recent results on systems of backward stochastic differential equations (BSDEs) with quadratic growth to the case of coupled forward-backward stochastic differential equations (FBSDEs). We work in a Markovian setting, and use results from the quadratic BSDE literature together with PDE techniques to obtain a-priori estimates which lead to an existence result. We also identify a general class of stochastic differential games whose corresponding FBSDE systems are covered by our main existence result. This leads to the existence of Markovian Nash equilibria for such games.

Linear-Quadratic-Gaussian mean-field controls of social optima

<p style='text-indent:20px;'>This paper investigates a class of unified stochastic linear-quadratic-Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a generalized setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be <i>indefinite</i> in cost functional. This setup is innovative and has great theoretical and realistic significance as its applications in mathematical finance (e.g., portfolio selection in mean-variation model). Using some <i>fully-coupled</i> variational analysis under the person-by-person optimality principle, and the mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with <i>embedding representation</i>. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.</p>

Stochastic Differential Equations with State Dependent Diffusion - 2 Order Statistics and State Estimation

Estimation of states in stochastic differential equations with state dependent diffusion is known to be difficult. Previous research recommend the higher order extended Kalman filter or the Lamperti transform method for this case. This paper shows that a new developed method, based on the unscented Kalman filter, is superior for two simulated stochastic differential equation systems.

Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II—the finite horizon case

We propose two numerical methods for the optimal control of McKean–Vlasov dynamics in finite time horizon. Both methods are based on the introduction of a suitable loss function defined over the parameters of a neural network. This allows the use of machine learning tools, and efficient implementations of stochastic gradient descent in order to perform the optimization. In the first method, the loss function stems directly from the optimal control problem. The second method tackles a generic forward-backward stochastic differential equation system (FBSDE) of McKean–Vlasov type, and relies on suitable reformulation as a mean field control problem. To provide a guarantee on how our numerical schemes approximate the solution of the original mean field control problem, we introduce a new optimization problem, directly amenable to numerical computation, and for which we rigorously provide an error rate. Several numerical examples are provided. Both methods can easily be applied to certain problems with common noise, which is not the case with the existing technology. Furthermore, although the first approach is designed for mean field control problems, the second is more general and can also be applied to the FBSDEs arising in the theory of mean field games.