152 research outputs found
On L_{p}-estimates of some singular integrals related to jump processes
We estimate fractional Sobolev and Besov norms of some singular integrals
arising in the model problem for the Zakai equation with discontinuous signal
and observation
On the rate of convergence of weak Euler approximation for non-degenerate SDEs
The paper estimates the rate of convergence of the weak Euler approximation
for the solutions of SDEs with Hoelder continuous coefficients driven by point
and martingale measures. The equation considered has a non-degenerate main part
whose jump intensity measure is absolutely continuous with respect to the Levy
measure of a spherically-symmetric stable process. It includes the
nondegenerate diffusions and SDEs driven by Levy processes.Comment: Added references, corrected typo
Model problem for integro-differential Zakai equation with discontinuous observation processes in H\"older spaces
The existence and uniqueness of solutions of the Cauchy problem to a a
stochastic parabolic integro-differential equation is investigated. The
equattion considered arises in nonlinear filtering problem with a jump signal
process and jump observation
On Lp -theory for parabolic and elliptic integro-differential equations with scalable operators in the whole space
Elliptic and parabolic integro-differential model problems are considered in
the whole space. By verifying H\"ormander condition, the existence and
uniqueness is proved in L_{p}-spaces of functions whose regularity is defined
by a scalable, possibly nonsymmetric, Levy measure. Some rough probability
density function estimates of the associated Levy process are used as well
On distribution free Skorokhod-Malliavin calculus
The starting point of the current paper is a sequence of uncorrelated random
variables. The distribution functions of these variables are assumed to be
given but no assumptions on the types or the structure of these distributions
are made. The above setting constitute the so called "distribution free"
paradigm. Under these assumptions, a version of Skorokhod-Malliavin calculus is
developed and applications to stochastic PDES are discussed
On the Cauchy problem for integro-differential operators in H\"older classes and the uniqueness of the martingale problem
The existence and uniqueness in H\"older spaces of solutions of the Cauchy
problem to parabolic integro-differential equation of the order
{\alpha}\in(0,2) is investigated. The principal part of the operator has kernel
m(t,x,y)/|y|^{d+{\alpha}} with a bounded nondegenerate m, H\"older in x and
measurable in y. The result is applied to prove the uniqueness of the
corresponding martingale problem
On the Cauchy problem for integro-differential equations in the scale of spaces of generalized smoothness
Parabolic integro-differential model Cauchy problem is considered in the
scale of Lp -spaces of functions whose regularity is defined by a scalable Levy
measure. Existence and uniqueness of a solution is proved by deriving apriori
estimates. Some rough probability density function estimates of the associated
Levy process are used as well
On the Cauchy problem for stochastic integro-differential equations with radially O-regularly varying Levy measure
Parabolic integro-differential nondegenerate Cauchy problem is considered in
the scale of L_{p} spaces of functions whose regularity is defined by a Levy
measure with O-regulary varying radial profile. Existence and uniqueness of a
solution is proved by deriving apriori estimates. Some probability density
function estimates of the associated Levy process are used as well.Comment: arXiv admin note: text overlap with arXiv:1805.0323
On Degenerate Linear Stochastic Evolution Equations Driven by Jump Processes
We prove the existence and uniqueness of solutions of degenerate linear
stochastic evolution equations driven by jump processes in a Hilbert scale
using the variational framework of stochastic evolution equations and the
method of vanishing viscosity. As an application of this result, we derive the
existence and uniqueness of solutions of degenerate parabolic linear stochastic
integro-differential equations (SIDEs) in the Sobolev scale. The SIDEs that we
consider arise in the theory of non-linear filtering as the equations governing
the conditional density of a degenerate jump-diffusion signal given a
jump-diffusion observation, possibly with correlated noise
On the Cauchy problem for nondegenerate parabolic integro-differential equations in the scale of generalized H\"older spaces
Parabolic integro-differential non degenerate Cauchy problem is considered in
the scale of H\"older spaces of functions whose regularity is defined by a
radially O-regularly varying L\'evy measure. Existence and uniqueness and the
estimates of the solution are derived
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