Type Stochastic Differential Equations Research Articles

An Approach to Studying Leontief Type Stochastic Differential Equations

An Approach to Studying Leontief Type Stochastic Differential Equations

Moderate deviations for rough differential equations

AbstractSmall noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter . We prove a moderate deviation principle for this equation as the scale parameter tends to zero.

Sliding Mode Fuzzy Control of Stochastic Nonlinear Systems Under Cyber-Attacks.

In this article, the problem of integral sliding mode control (ISMC) for a class of nonlinear systems with stochastic characteristics under cyber-attack is investigated. The control system and the cyber-attack are modeled as an Itô-type stochastic differential equation. The stochastic nonlinear systems are approached by the Takagi-Sugeno fuzzy model. A dynamic ISMC scheme is applied and the states and control input are analyzed within a universal dynamic model. It is demonstrated that trajectory of the system can be confined to the integral sliding surface within finite time, and the stability of closed-loop system under cyber-attack will be guaranteed by using a set of linear matrix inequalities. Following a standard procedure of universal fuzzy ISMC, it is shown that all signals in the closed-loop system will be guaranteed bounded, and the states are asymptotic stochastic stable if some conditions are met. An inverted pendulum is applied to show the effectiveness of our control scheme.

Solution theory to semilinear stochastic equations of Schrödinger type on curved spaces I: operators with uniformly bounded coefficients

We study the Cauchy problem for Schrödinger type stochastic semilinear partial differential equations with uniformly bounded variable coefficients, depending on the space variables. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.

Well-posedness and order preservation for neutral type stochastic differential equations of infinite delay with jumps

<abstract><p>In this work, we are concerned with the order preservation problem for multidimensional neutral type stochastic differential equations of infinite delay with jumps under non-Lipschitz conditions. By using a truncated Euler-Maruyama scheme and adopting an approximation argument, we have developed the well-posedness of solutions for a class of stochastic functional differential equations which allow the length of memory to be infinite, and the coefficients to be non-Lipschitz and even unbounded. Moreover, we have extended some existing conclusions on order preservation for stochastic systems to a more general case. A pair of examples have been constructed to demonstrate that the order preservation need not hold whenever the diffusion term contains a delay term, although the jump-diffusion coefficient could contain a delay term.</p></abstract>

Stability of Some Anticipating Semilinear Stochastic Differential Equations of Skorohod Type

AbstractIn the present paper, we study different types of stability of the solution of a semi-linear anticipating stochastic differential equation driven by a Brownian motion, with a random variable as initial condition. The involved stochastic integral is the Skorohod one. Being the initial condition random, we need to redefine the stability concepts. The new stability criteria depend on the derivative of the initial condition in the Malliavin calculus sense.

Wong–Zakai approximations for quasilinear systems of Itô's-type stochastic differential equations driven by fBm with H > 1 2

In a recent paper, Lanconelli and Scorolli 10 extended to the multidimensional case a Wong–Zakai-type approximation for Itô stochastic differential equations (SDEs) proposed by Øksendal and Hu 7 . 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 Hu 6 . 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.

Weak convergence of the backward Euler method for stochastic Cahn–Hilliard equation with additive noise

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen–Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn–Hilliard equation setting for the first time.

Interval neutrosophic stochastic dynamical systems driven by Brownian motion

Stochastic and neutrosophic theory are effective tools for modeling real-world phenomena and natural dynamical systems, where inputs are often affected by stochastic noises and outputs often contain both randomness and indeterminacy. In this work, we present a new type of stochastic differential equations (SDE) driven by an one-dimensional Brownian motion that can be considered as an efficient tool to describe the uncertain behavior of dynamical systems operating in interval neutrosophic environments with stochastic noises. After introducing some basic foundations on neutrosophic arithmetic, neutrosophic calculus and neutrosophic stochastic process, we define the new form of interval neutrosophic stochastic differential equations taking values in neutrosophic environment. Under some suitable conditions, the unique existence result of stochastic solution is obtained based on the use of Picard successive approximation. We also introduce an efficient numerical algorithm, namely Euler–Maruyama method, to solve the numerical solution of proposed problem and further demonstrate the effectiveness of the numerical method by solving some examples in stochastic biological systems such as stochastic logistic growth model, stochastic Lotka–Volterra predator–prey model, and stochastic SARS model, respectively.

A note about the invariance of the basic reproduction number for stochastically perturbed SIS models

AbstractIn Gray et al. [A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (3) (2011) 876–902] a susceptible‐infected‐susceptible (SIS) stochastic differential equation (SDE), obtained via a suitable random perturbation of the disease transmission coefficient in the classic SIS model, has been studied. Such random perturbation enters via an informal manipulation of stochastic differentials and leads to an Itô's‐type SDE. The authors identify a stochastic reproduction number, which differs from the standard one for the presence of those additional parameters that describe the employed random perturbation, and show that, similarly to the deterministic case, the stochastic reproduction number rules the asymptotic behavior of the solution. Aiming to make that random perturbation rigorous, we suggest an alternative approach based on a Wong–Zakai approximation argument thus arriving at a different stochastic model corresponding to the Stratonovich version of the Itô equation analyzed in Gray et al. Rather surprisingly, the asymptotic behavior of this alternative model turns out to be governed by the same reproduction number as the deterministic SIS equation. In other words, the random perturbation does not modify the threshold for extinction and persistence of the disease.

Parameter estimation for mean-reversion type stochastic differential equations from discrete observations

Parameter estimation for mean-reversion type stochastic differential equations from discrete observations

Parameter estimation for mean-reversion type stochastic differential equations from discrete observations

Parameter estimation for mean-reversion type stochastic differential equations from discrete observations

McKean\u2013Vlasov type stochastic differential equations arising from the random vortex method

We study a class of McKean–Vlasov type stochastic differential equations (SDEs) which arise from the random vortex dynamics and other physics models. By introducing a new approach we resolve the existence and uniqueness of both the weak and strong solutions for the McKean–Vlasov stochastic differential equations whose coefficients are defined in terms of singular integral kernels such as the Biot–Savart kernel. These SDEs which involve the distributions of solutions are in general not Lipschitz continuous with respect to the usual distances on the space of distributions such as the Wasserstein distance. Therefore there is an obstacle in adapting the ordinary SDE method for the study of this class of SDEs, and the conventional methods seem not appropriate for dealing with such distributional SDEs which appear in applications such as fluid mechanics.

Central Limit Theorem and Moderate Deviation Principle for McKean-Vlasov SDEs

Under a Lipschitz condition on distribution dependent coefficients, the central limit theorem and the moderate deviation principle are obtained for solutions of McKean-Vlasov type stochastic differential equations, which generalize the corresponding results for classical stochastic differential equations to the distribution dependent setting.

A higher order weak approximation of McKean–Vlasov type SDEs

The paper introduces a new weak approximation algorithm for stochastic differential equations (SDEs) of McKean–Vlasov type. The arbitrary order discretization scheme is available and is given using Malliavin weights, certain polynomial weights of Brownian motion, which play a role as correction of the approximation. The new weak approximation scheme works even if the test function is not smooth. In other words, the expectation of irregular functionals of McKean–Vlasov SDEs such as probability distribution functions are approximated through the proposed scheme. The effectiveness of the higher order scheme is confirmed by numerical examples for McKean–Vlasov SDEs including the Kuramoto model.

Wong–Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations

We extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Øksendal (1996) for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô’s SDEs, we consider for any finite partition of the time interval [0,T] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations 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 mesh of the partition tends to zero.

Effects of water currents on fish migration through a Feynman-type path integral approach under [Formula: see text] Liouville-like quantum gravity surfaces.

A stochastic differential game theoretic model has been proposed to determine optimal behavior of a fish while migrating against water currents both in rivers and oceans. Then, a dynamic objective function is maximized subject to two stochastic dynamics, one represents its location and another its relative velocity against water currents. In relative velocity stochastic dynamics, a Cucker-Smale type stochastic differential equation is introduced under white noise. As the information regarding hydrodynamic environment is incomplete and imperfect, a Feynman type path integral under [Formula: see text] Liouville-like quantum gravity surface has been introduced to obtain a Wick-rotated Schrödinger type equation to determine an optimal strategy of a fish during its migration. The advantage of having Feynman type path integral is that, it can be used in more generalized nonlinear stochastic differential equations where constructing a Hamiltonian-Jacobi-Bellman (HJB) equation is impossible. The mathematical analytic results show exact expression of an optimal strategy of a fish under imperfect information and uncertainty.

Non-equivalence of stochastic optimal control problems with open and closed loop controls

For an optimal control problem of an Itô’s type stochastic differential equation, the control process could be taken in open-loop or closed-loop forms. In the standard literature, provided appropriate regularity, the value functions under these two types of controls are equal and are the unique (viscosity) solution to the corresponding (path-dependent) HJB equation. In this short note, we provide a counterexample in the path dependent setting showing that these value functions can be different in general.

First and second order necessary optimality conditions for stochastic distributed systems

An optimal control problem of stochastic system, which that dynamics is described by hyperbolic type stochastic partial differential equations with a multipoint type quality criterion, is considered. A stochastic analog of the Euler equation is obtained and singular controls in the classical sense are investigated.

Stochastic averaging on a nonlinear oscillator with coordinate-dependent mass excited by Gaussian white noises

In this paper, a new stochastic averaging procedure is presented in detail to illustrate the improvement which develops the former stochastic averaging methods which can only deal with randomly excited oscillators without the coordinate-dependent mass (also called as inertia nonlinearity in many literature) into a new method which can handle nonlinear oscillators with both coordinate-dependent mass and stiffness nonlinearity terms. The crucial issue of this method is balancing the input and the dissipation of the Hamiltonian energy of the oscillator. After the oscillator has been simplified to be an I t ^ o type stochastic differential equation, the stationary probability density function (PDF) of transient equivalent amplitude, as well as the joint PDF of the displacement and velocity is studied. Numerical simulations are carried out to verify the theoretical anticipations. The numerical results coincide with the theoretical results perfectly.