We consider the stochastic differential equation (1) $du\left(t\right)=a(u\left(t\right),\xi \left(t\right))dt+{\int }_{\Theta }\sigma {(u\left(t\right),\theta )}_p(dt,d\theta )$ for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup ${P^t}_{t\ge 0}$ corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup ${P^t}_{t\ge 0}$ describing the evolution of measures along trajectories and vice versa.

In this paper, we introduce the Itô-Henstock integral of an operator-valued stochastic process and formulate a version of Itô's formula.

A new sufficient condition for the asymptotic stability of a locally Lipschitzian Markov semigroup acting on the space of signed measures ${}_{sig}$ is proved. This criterion is applied to the semigroup of Markov operators generated by a Poisson driven stochastic differential equation.

We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation $$d{X}_t^{\left(\alpha \right)}=-\frac{\alpha }{T-t}{X}_t^{\left(\alpha \right)}dt+d{B}_t,t\in [0,T)$$ , with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t)t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function...

This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale $N_t$ that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size $N_t$. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer...

We consider a stochastic process $X_t^x$ which solves an equation $$\mathrm{d}{X}_t^x=A{X}_t^x\mathrm{d}t+\Phi \mathrm{d}{B}_t^H,X_0^x=x$$ where $A$ and $\Phi$ are real matrices and $B^H$ is a fractional Brownian motion with Hurst parameter $H\in (1/2,1)$. The Kolmogorov backward equation for the function $u(t,x)=𝔼f\left(X_t^x\right)$ is derived and exponential convergence of probability distributions of solutions to the limit measure is established.

In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by ∫01X(s+i)/n dμ(s) for i=0, …, n−1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither markovian nor gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic...