We study a general stochastic rumour model in which an ignorant individual has a certain probability of becoming a stifler immediately upon hearing the rumour. We refer to this special kind of stifler as an uninterested individual. Our model also includes distinct rates for meetings between two spreaders in which both become stiflers or only one does, so that particular cases are the classical Daley-Kendall and Maki-Thompson models. We prove a Law of Large Numbers and a Central Limit Theorem for the proportions of those who ultimately remain ignorant and those who have heard the rumour but become uninterested in it.
We present two main theorems for stochastic processes with a Markov regime-switching model. First, we work on an existence-uniqueness theorem for a Stochastic Differential Delay Equation with Jumps and Regimes (SDDEJRs). Then, we provide the duality between an SDDEJR and an Anticipated Backward Stochastic Differential Equation with Jumps and Regimes (ABSDEJRs). Our goal is to provide two technical and basis theorems for the future theoretical and applied developments of time-delayed and time-advanced models.
This work concerns stochastic Volterra equations with singular kernels. Under the suitable conditions, we prove the central limit theorem for them. Moreover, we apply our result to stochastic Volterra equations with the kernels of fractional Brownian motions with the Hurst parameter $H\in(0, 1)$.
In this paper, we investigate suffcient and necessary conditions for the comparison theorem of neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDE). Moreover, the results extend the ones in the linear expectation case [1] and nonlinear expectation framework [8].
We prove a dual Yamada-Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) Levy processes. More precisely, we prove that weak uniqueness, i.e. uniqueness in law, implies weak joint uniqueness, i.e. joint uniqueness in law for the solution process and its driver.
We establish a central limit theorem and prove a moderate deviation principle for stochastic scalar conservation laws. Due to the lack of viscous term, this is done in the framework of kinetic solution. The weak convergence method and doubling variables method play a key role.
The work concerns invariant measures for multivalued McKean-Vlasov stochastic differential equations. First of all, we prove the exponential ergodicity of these equations. Then for a sequence of these equations, when their coefficients converge in the suitable sense, the convergence of corresponding strong solutions are presented. Finally, based on the convergence of these solutions, we establish the convergence of corresponding invariant measures.
We consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by cylindrical Brownian motion. The solutions are allowed to take values in general separable Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny's and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada-Watanabe theorem, i.e. we show that …
Fix $d\in\{1,2\}$, we consider a $d$-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang's condition. In this setting, we provide quantitative central limit theorems for the spatial average of the solution over a Euclidean ball, as the radius of the ball diverges to infinity. We also establish functional central limit theorems. A fundamental ingredient in our analysis is the pointwise $L^p$-estimate for the Malliavin derivative of the solution, which is of i…
We establish the large deviations principle (LDP) and the moderate deviations principle (MDP) and an almost sure version of the central limit theorem (CLT) for the stochastic 3D viscous primitive equations driven by a multiplicative white noise allowing dependence on spatial gradient of solutions with initial data in $H^2$. The LDP is established using the weak convergence approach of Budjihara and Dupuis and uniform version of the stochastic Gronwall lemma. The result corrects a minor technical issue in Z. Dong, J. Zhai, and R. Zhang: Large deviations principles for 3D stochastic primitive eq…
Dynkin's isomorphism theorem and the stochastic heat equation
考虑随机热方程$\partial_t u = \sL u + \dot{W}$，其中$\sL$是对偶中[Borel右]马尔可夫过程的生成元。我们表明，该解决方案是局部相互绝对连续的光滑扰动的高斯过程相关联的，通过Dynkin的同构定理，到本地时间的复制对称过程，对应于$\sL$ .在$\sL$的情况下，是一个发电机的Lévy过程$\R^d$，我们的结果给出了概率的解释，最近的研究结果Foondun等人。
