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We consider processes of the form [s,T]∋t↦u(t,X t ), where (X,P s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if $u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})$ with $\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})$ then there is a quasi-continuous version $\tilde{u}$ of u such that $\tilde{u}(t,X_{t})$ is a P s,x -Dirichlet process for quasi-every (s,x)∈[0,T)×ℝ d with respect to parabolic capacity, and we describe the martingale and the zero-quadratic variation parts of its decomposition. We also give conditions on u ensuring that $\tilde{u}(t,X_{t})$ is a semimartingale.

This paper concerns the dynamics of a stochastic SIVR epidemic model with imperfect vaccine where, differently from the epidemic model with perfect vaccine, the vaccinated is perturbed by the noise. This difference is the main difficulty to be conquered to give the threshold $R_{0}^{S}$ . Firstly, $R_{0}^{S}>1$ is proved to be sufficient for persistence in mean of the system. Then, three conditions for the disease to die out are given, which improve the previously known results on extinction of the disease. In case that the disease goes extinct, we show that the disease-free equilibrium is almost surely stable by using the nonnegative semimartingale convergence theorem.

In this paper we develop a class of applied probabilistic continuous time but discretized state space decompositions of the characterization of a multivariate generalized diffusion process. This decomposition is novel and, in particular, it allows one to construct families of mimicking classes of processes for such continuous state and continuous time diffusions in the form of a discrete state space but continuous time Markov chain representation. Furthermore, we present this novel decomposition and study its discretization properties from several perspectives. This class of decomposition both brings insight into understanding locally in the state space the induced dependence structures from the generalized diffusion process as well as admitting computationally efficient representations in order to evaluate functionals of generalized multivariate diffusion processes, which is based on a simple rank one tensor approximation of the exact representation. In particular, we investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale approach. We show that it is possible to represent, in both continuous and discrete space, that a multidimensional correlated generalized diffusion is a linear combination of processes originated from the decomposition of the starting multidimensional semimartingale. This result not only reconciles with the existing theory of diffusion approximations and decompositions, but defines the general representation of infinitesimal generators for both multidimensional generalized diffusions and, as we will demonstrate, also for the specification of copula density dependence structures. This new result provides immediate representation of the approximate weak solution for correlated stochastic differential equations. Finally, we demonstrate desirable convergence results for the proposed multidimensional semimartingales decomposition approximations.