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In 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. [ABSTRACT FROM AUTHOR]

We perturb with an additive noise the Hamiltonian system associated to a cubic anharmonic oscillator. This gives rise to a system of stochastic differential equations with quadratic drift and degenerate diffusion matrix. Firstly, we show that such systems possess explosive solutions for certain initial conditions. Then, we carry a small noise expansion's analysis of the stochastic system which is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We then investigate the probabilistic properties of the sequence of coefficients which turn out to be the unique strong solutions of stochastic perturbations of the well-known Lamé's equation. We also obtain explicit expressions of these in terms of Jacobi elliptic functions. Furthermore, we prove, in the case of Brownian noise, a lower bound for the probability that the truncated expansion stays close to the solution of the deterministic problem. Lastly, when the noise is bounded, we provide conditions for the almost sure convergence of the global expansion. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We link a general method for modeling random phenomena using systems of stochastic differential equations (SDEs) to the class of affine SDEs. This general construction emphasizes the central role of the Duffie–Kan system [Duffie and Kan, A yield-factor model of interest rates, Math. Finance 6 (1996) 379–406] as a model for first-order approximations of a wide class of nonlinear systems perturbed by noise. We also specialize to a two-dimensional framework and propose a direct proof of the Duffie–Kan theorem which does not passes through the comparison with an auxiliary process. Our proof produces a scheme to obtain an explicit representation of the solution once the one-dimensional square root process is assigned. [ABSTRACT FROM AUTHOR]

Inspired by the paper Greenhalgh et al. (Appl Math Comput 276:218–238, 2016), we investigate a class of two-dimensional stochastic differential equations related to susceptible–infected–susceptible epidemic models with demographic stochasticity. While preserving the key features of the model considered in Greenhalgh et al. (2016), where an ad hoc approach has been utilized to prove existence, uniqueness and non-explosivity of the solution, we consider an encompassing family of models described by a stochastic differential equation with random and Hölder continuous coefficients. We prove the existence of a unique strong solution by means of a Cauchy–Euler–Peano approximation scheme which is shown to converge in the proper topologies to the unique solution. [ABSTRACT FROM AUTHOR]

We prove that the asymptotic behavior of the Stratonovich counterpart of the Itô's type stochastic SIR model investigated in Tornatore et al. (2005), Ji and Jiang (2012, 2014) is ruled by the same threshold as the deterministic system. In other words, in contrast to the Itô's model, the intensity of the noise described through the Stratonovich calculus is not relevant for the extinction of the disease. The Stratonovich interpretation of the model is motivated by the parameter perturbation technique, employed on the disease transmission coefficient and used to implement environmental randomness, in combination with the classical Wong–Zakai approximation argument. [ABSTRACT FROM AUTHOR]

We prove a very general sharp inequality of the Hölder-Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong-Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Hölder inequality and Nelson's hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis. [ABSTRACT FROM AUTHOR]

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove an Itô-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, global Lipschitz continuity, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type. [Copyright &y& Elsevier]

We provide several inequalities for the Lq(P)-norm of the Wick product of random variables. These estimates are based on a Jensen's type inequality for the Wick multiplication, which we derive via a positivity argument. As an application we study a certain type of anticipating stochastic differential equation whose solution is shown to be an element of Lq(P) for some q ≥ 1. [ABSTRACT FROM AUTHOR]

In this study we investigate a novel approach to stochastically perturb the disease transmission coefficient, which is a key parameter in susceptible-infected-susceptible (SIS) models. Motivated by the papers [5] and [2], we perturb the disease transmission coefficient with a Gaussian white noise, formally modelled as the time derivative of a mean reverting Ornstein-Uhlenbeck process. We remark that, thanks to a suitable representation of the solution to the deterministic SIS model, this perturbation is rigorous and supported by a Wong-Zakai approximation argument that consists in smoothing the singular Gaussian white noise and then taking limit of the solution from the approximated model. We prove that the stochastic version of the classic SIS model obtained this way preserves a crucial feature of the deterministic equation: the reproduction number dictating the two possible asymptotic regimes for the infection, i.e. extinction and persistence, remains unchanged. We then identify the class of perturbing noises for which this property holds and propose simple sufficient conditions for that. All the theoretical discoveries are illustrated and discussed with the help of several numerical simulations. [ABSTRACT FROM AUTHOR]

The concept of Wick product is strongly related to the underlying Brownian motion we have fixed on the probability space. Via the Girsanov's theorem we construct a family of new Brownian motions, obtained as translations of the original one, and to each of them we associate a Wick product. This produces a family of Wick products, named γ-Wick products, parameterized by the performed translations. We aim to describe this family of products. We also define a new family of stochastic integrals, which are related in a natural way to the γ-Wick products. [ABSTRACT FROM AUTHOR]

We study multidimensional diffusion processes and give an explicit representation for their conditional expectation. Starting from the solution formula for one dimensional stochastic differential equations found in Lanconelli and Proske [8], we compute the conditional expectation of a certain class of multidimensional diffusions without resorting to the Markov property of the process and therefore without requiring an explicit expression for the semi group associated to it. [ABSTRACT FROM AUTHOR]

We investigate the regularity of the law of Wong-Zakai-type approximations for Itô stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using criteria based on the Malliavin calculus we establish absolute continuity and a Fokker-Planck-type equation solved in the distributional sense by the density. The parabolic smoothing effect typical of the solutions of Itô equations is lacking in this approximated framework; therefore, in order to prove absolute continuity, the initial condition of the random differential equation needs to possess a density itself. [ABSTRACT FROM AUTHOR]