Related papers: Random time changes of Feller processes

Let $(L_t)_{t \geq 0}$ be a $k$-dimensional L\'evy process and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times k}$ a continuous function such that the L\'evy-driven stochastic differential equation (SDE) $$dX_t = \sigma(X_{t-}) \, dL_t,…

We consider the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $\alpha$-stable process $Z_t$ in $R^d$, where $\alpha \in (0,1)$ and $d \ge 2$. We assume that the determinant of $A(x) =…

A time and space inhomogeneous Markov process is a Feller evolution process, if the corresponding evolution system on the continuous functions vanishing at infinity is strongly continuous. We discuss generators of such systems and show that…

In this paper, we consider a long-time behavior of stable-like processes. A stable-like process is a Feller process given by the symbol $p(x,\xi)=-i\beta(x)\xi+\gamma(x)|\xi|^{\alpha(x)},$ where $\alpha(x)\in(0,2)$, $\beta(x)\in\R$ and…

Let $\alpha\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $\alpha$-stable process in $\mathbb{R}^d$: $$ dX_t=b(X_t)dt+\sigma(X_{t-})d L^{\alpha}_t, \quad X_0=x\in\mathbb{R}^d, $$…

In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\bf R}^d$: $$ d X_{t}= \sigma_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^d, $$ where $Z$ is a $d$-dimensioanl nondegenerate…

Let $(U_t,V_t)$ be a bivariate L\'evy process, where $V_t$ is a subordinator and $U_t$ is a L\'evy process formed by randomly weighting each jump of $V_t$ by an independent random variable $X_t$ having cdf $F$. We investigate the asymptotic…

We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for L\'{e}vy processes. The proof uses a…

Let $u$ be the solution to the following stochastic evolution equation (1) du(t,x)& = &A u(t,x) dt + B \sigma(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking values in an Hilbert space $\HH$, where $L$ is a $\RR$ valued L\'evy process, $A:H\to…

In this paper we study the domain of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of L\'evy- and L\'evy-type…

We consider solutions of L\'evy-driven stochastic differential equations of the form $\mathrm{d} X_t=\sigma(X_{t-})\mathrm{d} L_t$, $X_0=x$ where the function $\sigma$ is twice continuously differentiable and maximal of linear growth and…

We show existence of an infinitesimally invariant measure $m$ for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some…

We study asymptotic behavior of maximum likelihood estimator for a time inhomogeneous diffusion process given by a SDE $dX_t=\alpha b(t)X_t dt + \sigma(t) dB_t$, $t\in[0,T)$, with a parameter $\alpha\in R$, where $T\in(0,\infty]$ and…

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem implies that the unique…

We consider stochastic differential equations (SDEs) driven by Feller processes which are themselves solutions of multivariate Levy driven SDEs. The solutions of these 'iterated SDEs' are shown to be non-Markovian. However, the process…

In this paper, we study the following supercritical McKean-Vlasov SDE, driven by a symmetric non-degenerate cylindrical $\alpha$-stable process in $\mathbb{R}^d$ with $\alpha \in (0,1)$: $$ \mathord{{\rm d}} X_t = (K *…

Let $d \ge 2$. In this paper, we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dS_{t}+b(s+t, X_{t})dt, \quad X_{0}=x, \] where $(s,x)\in \mathbb{R}_+ \times \mathbb{R}^{d}$ is the initial starting…

We consider the exponential functional $A_{\infty}=\int_0^{\infty} e^{\xi_s} ds$ associated to a Levy process $(\xi_t)_{t \geq 0}$. We find the asymptotic behavior of the tail of this random variable, under some assumptions on the process…

In this article, we consider additive functionals $\zeta_t = \int_0^t f(X_s)\mathrm{d} s$ of a c\`adl\`ag Markov process $(X_t)_{t\geq 0}$ on $\mathbb{R}$. Under some general conditions on the process $(X_t)_{t\geq 0}$ and on the function…

Let $X_{\alpha}=\{X_{\alpha}(t),t\in T\}$, $\alpha>0$, be an $\alpha$-permanental process with kernel $u(s,t)$. We show that $X^{1/2}_{\alpha}$ is a subgaussian process with respect to the metric $\sigma (s,t)=…