187 research outputs found
Finite Difference Schemes for Stochastic Partial Differential Equations in Sobolev Spaces
We discuss -estimates for finite difference schemes approximating
parabolic, possibly degenerate, SPDEs, with initial conditions from and
free terms taking values in Consequences of these estimates include an
asymptotic expansion of the error, allowing the acceleration of the
approximation by Richardson's method.Comment: 22 pages. The final publication is available at Springer via
http://dx.doi.org/10.1007/s00245-014-9272-
On the solvability of degenerate stochastic partial differential equations in Sobolev spaces
Systems of parabolic, possibly degenerate parabolic SPDEs are considered.
Existence and uniqueness are established in Sobolev spaces. Similar results are
obtained for a class of equations generalizing the deterministic first order
symmetric hyperbolic systems.Comment: 26 page
Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion
In this paper, we consider a product of a symmetric stable process in
and a one-dimensional Brownian motion in . Then we
define a class of harmonic functions with respect to this product process. We
show that bounded non-negative harmonic functions in the upper-half space
satisfy Harnack inequality and prove that they are locally H\"older continuous.
We also argue a result on Littlewood-Paley functions which are obtained by the
-harmonic extension of an function.Comment: 23 page
The filtering equations revisited
The problem of nonlinear filtering has engendered a surprising number of
mathematical techniques for its treatment. A notable example is the
change-of--probability-measure method originally introduced by Kallianpur and
Striebel to derive the filtering equations and the Bayes-like formula that
bears their names. More recent work, however, has generally preferred other
methods. In this paper, we reconsider the change-of-measure approach to the
derivation of the filtering equations and show that many of the technical
conditions present in previous work can be relaxed. The filtering equations are
established for general Markov signal processes that can be described by a
martingale-problem formulation. Two specific applications are treated
Mutual Fund Theorem for continuous time markets with random coefficients
We study the optimal investment problem for a continuous time incomplete
market model such that the risk-free rate, the appreciation rates and the
volatility of the stocks are all random; they are assumed to be independent
from the driving Brownian motion, and they are supposed to be currently
observable. It is shown that some weakened version of Mutual Fund Theorem holds
for this market for general class of utilities; more precisely, it is shown
that the supremum of expected utilities can be achieved on a sequence of
strategies with a certain distribution of risky assets that does not depend on
risk preferences described by different utilities.Comment: 17 page
Frazil ice formation during the spring flood and its role in transport of sediments to the ice cover
Maximal regularity for non-autonomous equations with measurable dependence on time
In this paper we study maximal -regularity for evolution equations with
time-dependent operators . We merely assume a measurable dependence on time.
In the first part of the paper we present a new sufficient condition for the
-boundedness of a class of vector-valued singular integrals which does not
rely on H\"ormander conditions in the time variable. This is then used to
develop an abstract operator-theoretic approach to maximal regularity.
The results are applied to the case of -th order elliptic operators
with time and space-dependent coefficients. Here the highest order coefficients
are assumed to be measurable in time and continuous in the space variables.
This results in an -theory for such equations for .
In the final section we extend a well-posedness result for quasilinear
equations to the time-dependent setting. Here we give an example of a nonlinear
parabolic PDE to which the result can be applied.Comment: Application to a quasilinear equation added. Accepted for publication
in Potential Analysi
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