1,747 research outputs found

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

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    We prove It\^o's formula for the LpL_{p}-norm of a stochastic Wp1W^{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

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    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 ppth power, pβ‰₯2p\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

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    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

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    We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order h1/2h^{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

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    We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of LpL_{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\gammath norm of the second-order derivatives, where γ∈(0,1)\gamma\in(0,1), and the second type deals with estimates of the resolvent operators in LpL_{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

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    We establish the existence of solutions of fully nonlinear parabolic second-order equations like βˆ‚tu+H(v,Dv,D2v,t,x)=0\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0 in smooth cylinders without requiring HH to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of HH at points at which ∣D2vβˆ£β‰€K|D^{2}v|\leq K, where KK is any fixed constant. For large ∣D2v∣|D^{2}v| some kind of relaxed convexity assumption with respect to D2vD^{2}v mixed with a VMO condition with respect to t,xt,x are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on HH, apart from ellipticity, but of a "cut-off" version of the equation βˆ‚tu+H(v,Dv,D2v,t,x)=0\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

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    We prove several pointwise estimates for solutions of linear elliptic (parabolic) equations with measurable coefficients in smooth domains (cylinders) through the weighted LdL_{d} (Ld+1L_{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

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    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

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    We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman's equations is of order at least h2/3h^{2/3}, where hh is the mesh size. The equations are considered in smooth bounded domains.Comment: 24 page

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

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    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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