Related papers: Cylindrical Martingale Problems Associated with L\…
We prove limit theorems for cylindrical martingale problems associated to L\'evy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss…
Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and…
Let $A$ be a pseudo-differential operator with symbol $q(x,\xi)$. In this paper we derive sufficient conditions which ensure the existence of a solution to the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem. If the symbol $q$ depends…
We introduce a stochastic integral with respect to cylindrical L\'evy processes with finite $p$-th weak moment for $p\in [1,2]$. The space of integrands consists of $p$-summing operators between Banach spaces of martingale type $p$. We…
We consider systems of stochastic differential equations of the form \[ \d X_t^i = \sum_{j=1}^d A_{ij}(X_{t-}) \d Z_t^j\] for $i=1,\dots,d$ with continuous, bounded and non-degenerate coefficients. Here $Z_t^1,\dots,Z_t^d$ are independent…
In this paper, we deal with a class of reflected backward stochastic differential equations associated to the subdifferential operator of a lower semi-continuous convex function driven by Teugels martingales associated with L\'{e}vy…
This paper considers the martingale problem for a class of weakly coupled L\'{e}vy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process…
In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Levy process. Our approach requires to establish a…
In this article we study a class of stochastic functional differential equations driven by L\'{e}vy processes (in particular, $\alpha$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals.…
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 obtain Calder{\'o}n-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous coefficients. We then derive the well-posedness of the martingale problem associated to related degenerate operators, and therefore…
The problem of finding a martingale on a manifold with a fixed random terminal value can be solved by considering BSDEs with a generator with quadratic growth. We study here a generalization of these equations and we give uniqueness and…
We prove that weakly continuous solutions to martingale problems admit a canonical regular conditional probability distribution. This allows for the construction of time consistent convex dynamic procedures in a non dominated setting.…
We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, $L^p$ data and generators satisfying weak integrability conditions. We deal with equations…
We use the martingale convergence method to get the weak convergence theorem on general functionals of partial sums of independent heavy-tailed random variables. The limiting process is the stochastic integral driven by $\alpha-$stable…
In this paper, several weak Orlicz-Hardy martingale spaces associated with concave functions are introduced, and some weak atomic decomposition theorems for them are established. With the help of weak atomic decompositions, a sufficient…
In this work, by using Levi's parametrix method we first construct the fundamental solution of the critical non-local operator perturbed by gradient. Then, we use the obtained estimates to prove the existence and uniqueness of strong…
Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and…
In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical L\'evy process, and show that these conditions are also necessary if the…
In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an…