126 research outputs found
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 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 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
Global L_2-solutions of stochastic Navier-Stokes equations
This paper concerns the Cauchy problem in R^d for the stochastic
Navier-Stokes equation \partial_tu=\Delta u-(u,\nabla)u-\nabla p+f(u)+
[(\sigma,\nabla)u-\nabla \tilde p+g(u)]\circ \dot W, u(0)=u_0,\qquad divu=0,
driven by white noise \dot W. Under minimal assumptions on regularity of the
coefficients and random forces, the existence of a global weak (martingale)
solution of the stochastic Navier-Stokes equation is proved. In the
two-dimensional case, the existence and pathwise uniqueness of a global strong
solution is shown. A Wiener chaos-based criterion for the existence and
uniqueness of a strong global solution of the Navier-Stokes equations is
established.Comment: Published at http://dx.doi.org/10.1214/009117904000000630 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On Some Properties of Space Inverses of Stochastic Flows
We derive moment estimates and a strong limit theorem for space inverses of
stochastic flows generated by jump SDEs with adapted coefficients in weighted
H\"older norms using the Sobolev embedding theorem and the change of variable
formula. As an application of some basic properties of flows of continuous
SDEs, we derive the existence and uniqueness of classical solutions of linear
parabolic second order SPDEs by partitioning the time interval and passing to
the limit. The methods we use allow us to improve on previously known results
in the continuous case and to derive new ones in the jump case.Comment: 30 pages; Part of the material of this paper is from the first
version of our paper entitled "On Classical Solutions of Linear Stochastic
Integro-Differential Equations" (arXiv:1404.0345). We have removed this
material from arXiv:1404.034
On Classical Solutions of Linear Stochastic Integro-Differential Equations
We prove the existence of classical solutions to parabolic linear stochastic
integro-differential equations with adapted coefficients using Feynman-Kac
transformations, conditioning, and the interlacing of space-inverses of
stochastic flows associated with the equations. The equations are forward and
the derivation of existence does not use the "general theory" of SPDEs.
Uniqueness is proved in the class of classical solutions with polynomial
growth.Comment: 50 pages; We have removed some of the material on inverse flows and
moved it to the paper "On Some Properties of Space Inverses of Stochastic
Flows" (arXiv:1411.6277). Also, the assumptions for our main existence
theorem (Theorem 2.5 in new version) have been modified and we have
formulated our representation theorem (Theorem 2.2 in new version) for an
equation with a special for
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