Related papers: The martingale problem for geometric stable-like p…

In this paper we introduce the concept of conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about…

We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel. This class of processes contains, for example, fractional L\'{e}vy processes as…

Linear multivariate Hawkes processes (MHP) are a fundamental class of point processes with self-excitation. When estimating parameters for these processes, a difficulty is that the two main error functionals, the log-likelihood and the…

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative…

This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a…

Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $\pi$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to…

We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study…

Additive processes are obtained from L\'{e}vy ones by relaxing the condition of stationary increments, hence they are spatially (but not temporally) homogeneous. By analogy with the case of time-homogeneous Markov processes, one can define…

For two Polish state spaces $E_X$ and $E_Y$, and an operator $G_X$, we obtain existence and uniqueness of a $G_X$-martingale problem provided there is a bounded continuous duality function $H$ on $E_X \times E_Y$ together with a dual…

Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than a decade. One of the most well-known and widely studied problems is that of estimation of the quadratic…

We present simple new examples of pure-jump strict local martingales. The examples are constructed as exponentials of self-exciting affine Markov processes. We characterize the strict local martingale property of these processes by an…

Let $A_\pm>0$, $\beta\in(0,1)$, and let $Z^{(\alpha)}$ be a strictly $\alpha$-stable L\'evy process with the jump measure $\nu(\mathrm{d} z)=(C_+\mathbb{I}_{(0,\infty)}(z)+ C_-\mathbb{I}_{(-\infty,0)}(z))|z|^{-1-\alpha}\,\mathrm{d} z$,…

Using complex analysis techniques we obtain precise asymptotic approximations for the kernels corresponding to the symmetric $\alpha$-stable processes and their fractional derivatives. We apply our method to general L\'evy processes whose…

For a discrete-time Markov chain $\{X(t)\}$ evolving on $\Re^\ell$ with transition kernel $P$, natural, general conditions are developed under which the following are established: 1. The transition kernel $P$ has a purely discrete spectrum,…

Pure-jump L\'evy processes are popular classes of stochastic processes which have found many applications in finance, statistics or machine learning. In this paper, we propose a novel family of self-decomposable L\'evy processes where one…

Using key tools such as It\^o formula for general semi-martingales, moments estimates for L\'{e}vy-type stochastic integrals and properties of regular varying functions we find conditions under which solutions of stochastic differential…

We present an existence result for L\'evy-type processes which requires only weak regularity assumptions on the symbol $q(x,\xi)$ with respect to the space variable $x$. Applications range from existence and uniqueness results for…

In this paper we show the H\"ormander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general H\"ormander's Lie bracket conditions, we show the regularization…

Given a stable L\'{e}vy process $X=(X_t)_{0\le t\le T}$ of index $\alpha\in(1,2)$ with no negative jumps, and letting $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t\in [0,T]$, we consider the optimal prediction problem…

In this paper, we study the transition densities of pure-jump symmetric Markov processes in $ {{\mathbb R}}^d$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions…