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N. V. Krylov

Publications and source records attributed to N. V. Krylov.

At least 91 records · Page 5Linked to original sources
arXiv2012

On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains

We prove the dynamic programming principe for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. The zeroth-order "coefficient" and the "free" term are only assumed to be measurable. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. The main assumption, which will be removed in a subsequent article, is that there exists a sufficiently regular solution of the Isaacs equation.

math.OC↗
arXiv2012

On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the Isaacs equations

We prove the dynamic programming principe for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order "coefficient" and the "free" term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions.

math.OC↗
arXiv2012

On the existence of smooth solutions for fully nonlinear elliptic equations with measurable "coefficients" without convexity assumptions

We show that for any uniformly elliptic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term one can find an approximating equation which has a unique continuous and having the second derivatives locally bounded solution in a given smooth domain with smooth boundary data. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its derivatives.

math.AP↗
arXiv2012

On the existence of $W^{2}_{p}$ solutions for fully nonlinear elliptic equations under relaxed convexity assumptions

We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of $H$ at points at which $|D^{2}v|\leq K$, where $K$ is any given constant. For large $|D^{2}v|$ some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a VMO condition with respect to $x$ are still imposed. The solutions are sought in Sobolev classes.

math.AP↗
arXiv2012

Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients

We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order equations in \cite{DKL}. The first type is an estimate of the $γ$th norm of the second-order derivatives, where $γ\in(0,1)$, and the second type deals with estimates of the resolvent operators in $L_{p}$ when the first-order coefficients are summable to an appropriate power.

math.AP↗
arXiv2010

On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients

We prove the solvability in Sobolev spaces $W^{1,2}_p$, $p>d+1$, of the terminal-boundary value problem for a class of fully nonlinear parabolic equations, including parabolic Bellman's equations, in bounded cylindrical domains with VMO ``coefficients''. The solvability in $W^{2}_p$, $p>d$, of the corresponding elliptic boundary-value problem is also obtained.

math.AP↗
arXiv2010

Kalman-Bucy filter and SPDEs with growing lower-order coefficients in $W^{1}_{p}$ spaces without weights

We consider divergence form uniformly parabolic SPDEs with VMO bounded leading coefficients, bounded coefficients in the stochastic part, and possibly growing lower-order coefficients in the deterministic part. We look for solutions which are summable to the $p$th power, $p\geq2$, with respect to the usual Lebesgue measure along with their first-order derivatives with respect to the spatial variable. Our methods allow us to include Zakai's equation for the Kalman-Bucy filter into the general filtering theory.

math.AP↗
arXiv2010

On Bellman's equations with VMO coefficients

We present a result about solvability in $W^{2}_{p}$, $p>d$, in the whole space $\bR^{d}$ of Bellman's equations with VMO ``coefficients''. Parabolic equations are touched upon as well.

math.AP↗
arXiv2009

Second-order elliptic and parabolic equations with $B(\mathbb R^{2}, VMO)$ coefficients

The solvability in Sobolev spaces $W^{1,2}_p$ is proved for nondivergence form second order parabolic equations for $p>2$ close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the $W^{2}_p$-solvability for the same $p$ of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when $p=2$.

math.AP↗
arXiv2009

On divergence form SPDEs with growing coefficients in $W^{1}_{2}$ spaces without weights

We consider divergence form uniformly parabolic SPDEs with bounded and measurable leading coefficients and possibly growing lower-order coefficients in the deterministic part of the equations. We look for solutions which are summable to the second power with respect to the usual Lebesgue measure along with their first derivatives with respect to the spatial variable.

math.PR↗
arXiv2009

SPDEs in divergence form with VMO coefficients and filtering theory of partially observable diffusion processes with Lipschitz coefficients

We present several results on the smoothness in $L_{p}$ sense of filtering densities under the Lipschitz continuity assumption on the coefficients of a partially observable diffusion processes. We obtain them by rewriting in divergence form filtering equation which are usually considered in terms of formally adjoint to operators in nondivergence form.

math.PR↗
arXiv2008

Second-order elliptic equations with variably partially VMO coefficients

The solvability in $W^{2}_{p}(\bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This generalizes to a very large extent the case of equations with continuous or VMO coefficients.

math.AP↗