Displaying similar documents to “Strong solutions for stochastic differential equations with jumps”

Uniqueness for stochastic evolution equations in Banach spaces

Martin Ondreját


Different types of uniqueness (e.g. pathwise uniqueness, uniqueness in law, joint uniqueness in law) and existence (e.g. strong solution, martingale solution) for stochastic evolution equations driven by a Wiener process are studied and compared. We show a sufficient condition for a joint distribution of a process and a Wiener process to be a solution of a given SPDE. Equivalences between different concepts of solution are shown. An alternative approach to the construction of the stochastic...

Tightness of Continuous Stochastic Processes

Michał Kisielewicz (2006)

Discussiones Mathematicae Probability and Statistics


Some sufficient conditins for tightness of continuous stochastic processes is given. It is verified that in the classical tightness sufficient conditions for continuous stochastic processes it is possible to take a continuous nondecreasing stochastic process instead of a deterministic function one.

Stochastic differential inclusions

Michał Kisielewicz (1997)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization


The definition and some existence theorems for stochastic differential inclusions depending only on selections theorems are given.

A second order SDE for the Langevin process reflected at a completely inelastic boundary

Jean Bertoin (2008)

Journal of the European Mathematical Society


It was shown in [2] that a Langevin process can be reflected at an energy absorbing boundary. Here, we establish that the law of this reflecting process can be characterized as the unique weak solution to a certain second order stochastic differential equation with constraints, which is in sharp contrast with a deterministic analog.

Some applications of Girsanov's theorem to the theory of stochastic differential inclusions

Micha Kisielewicz (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization


The Girsanov's theorem is useful as well in the general theory of stochastic analysis as well in its applications. We show here that it can be also applied to the theory of stochastic differential inclusions. In particular, we obtain some special properties of sets of weak solutions to some type of these inclusions.

Splitting the conservation process into creation and annihilation parts

Nicolas Privault (1998)

Banach Center Publications


The aim of this paper is the study of a non-commutative decomposition of the conservation process in quantum stochastic calculus. The probabilistic interpretation of this decomposition uses time changes, in contrast to the spatial shifts used in the interpretation of the creation and annihilation operators on Fock space.

Equations in differentials in the algebra of generalized stochastic processes

Nadzeya V. Bedziuk, Aleh L. Yablonski (2010)

Banach Center Publications


We consider an ordinary or stochastic nonlinear equation with generalized coefficients as an equation in differentials in the algebra of new generalized functions in the sense of [8]. Consequently, the solution of such an equation is a new generalized function. We formulate conditions under which the solution of a given equation in the algebra of new generalized functions is associated with an ordinary function or process. Moreover the class of all possible associated functions and processes...

SPDEs with pseudodifferential generators: the existence of a density

Samy Tindel (2000)

Applicationes Mathematicae


We consider the equation du(t,x)=Lu(t,x)+b(u(t,x))dtdx+σ(u(t,x))dW(t,x) where t belongs to a real interval [0,T], x belongs to an open (not necessarily bounded) domain 𝒪 , and L is a pseudodifferential operator. We show that under sufficient smoothness and nondegeneracy conditions on L, the law of the solution u(t,x) at a fixed point ( t , x ) [ 0 , T ] × 𝒪 is absolutely continuous with respect to the Lebesgue measure.