It\^o's formula for the LpL_{p}-norm of stochastic Wp1W^{1}_{p}-valued processes

We prove It\^o's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.Comment: 16 page

Kalman-Bucy filter and SPDEs with growing lower-order coefficients in Wp1W^{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.Comment: 43 page

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.Comment: 29 pages. Few inconsistencies and misprints corrected, two references adde

On the rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients

We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order $h^{1/2}$ for certain types of finite-difference schemes are obtained.Comment: 32 page

Some LpL_{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 $\gamma$th norm of the second-order derivatives, where $\gamma\in(0,1)$, 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.Comment: 23 pages, submitte

On the existence of Wp1,2W^{1,2}_{p} solutions for fully nonlinear parabolic equations under either relaxed or no convexity assumptions

We establish the existence of solutions of fully nonlinear parabolic second-order equations like $\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0$ in smooth cylinders 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 fixed 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 $t,x$ are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on $H$, apart from ellipticity, but of a "cut-off" version of the equation $\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0$.Comment: 30 pages, a few errors correcte

Weighted Aleksandrov estimates: PDE and stochastic versions

We prove several pointwise estimates for solutions of linear elliptic (parabolic) equations with measurable coefficients in smooth domains (cylinders) through the weighted $L_{d}$ ($L_{d+1}$)-norm of the free term. The weights allow the free term to blow up near the (latteral) boundary. We also present weighted estimates for occupation times of diffusion processes.Comment: 27 page

On the paper "All functions are locally s-harmonic up to a small error" by Dipierro, Savin, and Valdinoci

We give an appropriate version of the result in the paper by Dipierro, Savin, and Valdinoci for different, not necessarily fractional, powers of the Laplacian.Comment: 4 page

Rate of convergence of difference approximations for uniformly nondegenerate elliptic Bellman's equations

We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman's equations is of order at least $h^{2/3}$, where $h$ is the mesh size. The equations are considered in smooth bounded domains.Comment: 24 page

H\"ormander's theorem for stochastic partial differential equations

We prove H\"ormander's type hypoellipticity theorem for stochastic partial differential equations when the coefficients are only measurable with respect to the time variable. The need for such kind of results comes from filtering theory of partially observable diffusion processes, when even if the initial system is autonomous, the observation process enters the coefficients of the filtering equation and makes them time-dependent with no good control on the smoothness of the coefficients with respect to the time variable.Comment: 23 pages, localization on random events adde