2,218 research outputs found
A regularity result for quasilinear stochastic partial differential equations of parabolic type
We consider a quasilinear parabolic stochastic partial differential equation
driven by a multiplicative noise and study regularity properties of its weak
solution satisfying classical a priori estimates. In particular, we determine
conditions on coefficients and initial data under which the weak solution is
H\"older continuous in time and possesses spatial regularity that is only
limited by the regularity of the given data. Our proof is based on an efficient
method of increasing regularity: the solution is rewritten as the sum of two
processes, one solves a linear parabolic SPDE with the same noise term as the
original model problem whereas the other solves a linear parabolic PDE with
random coefficients. This way, the required regularity can be achieved by
repeatedly making use of known techniques for stochastic convolutions and
deterministic PDEs
Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs
We study an expansion method for high-dimensional parabolic PDEs which
constructs accurate approximate solutions by decomposition into solutions to
lower-dimensional PDEs, and which is particularly effective if there are a low
number of dominant principal components. The focus of the present article is
the derivation of sharp error bounds for the constant coefficient case and a
first and second order approximation. We give a precise characterisation when
these bounds hold for (non-smooth) option pricing applications and provide
numerical results demonstrating that the practically observed convergence speed
is in agreement with the theoretical predictions
Some regularity and convergence results for parabolic Hamilton-Jacobi-Bellman equations in bounded domains
We study the approximation of parabolic Hamilton-Jacobi-Bellman (HJB)
equations in bounded domains with strong Dirichlet boundary conditions. We work
under the assumption of the existence of a sufficiently regular barrier
function for the problem to obtain well-posedness and regularity of a related
switching system and the convergence of its components to the HJB equation. In
particular, we show existence of a viscosity solution to the switching system
by a novel construction of sub- and supersolutions and application of Perron's
method. Error bounds for monotone schemes for the HJB equation are then derived
from estimates near the boundary, where the standard regularisation procedure
for viscosity solutions is not applicable, and are found to be of the same
order as known results for the whole space. We deduce error bounds for some
common finite difference and truncated semi-Lagrangian schemes
Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise
In this article, we are concerned with a multidimensional degenerate
parabolic-hyperbolic equation driven by Levy processes. Using bounded variation
(BV) estimates for vanishing viscosity approximations, we derive an explicit
continuous dependence estimate on the nonlinearities of the entropy solutions
under the assumption that Levy noise depends only on the solution. This result
is used to show the error estimate for the stochastic vanishing viscosity
method. In addition, we establish fractional BV estimate for vanishing
viscosity approximations in case the noise coefficients depend on both the
solution and spatial variable.Comment: 31 Pages. arXiv admin note: text overlap with arXiv:1502.0249
Intrinsic expansions for averaged diffusion processes
We show that the rate of convergence of asymptotic expansions for solutions
of SDEs is generally higher in the case of degenerate (or partial) diffusion
compared to the elliptic case, i.e. it is higher when the Brownian motion
directly acts only on some components of the diffusion. In the scalar case,
this phenomenon was already observed in (Gobet and Miri 2014) using Malliavin
calculus techniques. In this paper, we provide a general and detailed analysis
by employing the recent study of intrinsic functional spaces related to
hypoelliptic Kolmogorov operators in (Pagliarani et al. 2016). Relevant
applications to finance are discussed, in particular in the study of
path-dependent derivatives (e.g. Asian options) and in models incorporating
dependence on past information
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