2,366 research outputs found

### It\^o's formula for the $L_{p}$-norm of stochastic $W^{1}_{p}$-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 $W^{1}_{p}$ 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.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 $L_{p}$-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 $\gamma$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 $W^{1,2}_{p}$ 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

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