67 research outputs found
Solving linear parabolic rough partial differential equations
We study linear rough partial differential equations in the setting of [Friz
and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear
parabolic partial differential equation driven by a deterministic rough path
of H\"older regularity with . Based on a stochastic representation of the solution of the rough
partial differential equation, we propose a regression Monte Carlo algorithm
for spatio-temporal approximation of the solution. We provide a full
convergence analysis of the proposed approximation method which essentially
relies on the new bounds for the higher order derivatives of the solution in
space. Finally, a comprehensive simulation study showing the applicability of
the proposed algorithm is presented
On the splitting-up method for rough (partial) differential equations
This article introduces the splitting method to systems responding to rough
paths as external stimuli. The focus is on nonlinear partial differential
equations with rough noise but we also cover rough differential equations.
Applications to stochastic partial differential equations arising in control
theory and nonlinear filtering are given
An energy method for rough partial differential equations
Hocquet A, Hofmanová M. An energy method for rough partial differential equations. JOURNAL OF DIFFERENTIAL EQUATIONS. 2018;265(4):1407-1466.We present a well-posedness and stability result for a class of nondegenerate linear parabolic equations driven by geometric rough paths. More precisely, we introduce a notion of weak solution that satisfies an intrinsic formulation of the equation in a suitable Sobolev space of negative order. Weak solutions are then shown to satisfy the corresponding energy estimates which are deduced directly from the equation. Existence is obtained by showing compactness of a suitable sequence of approximate solutions whereas uniqueness relies on a doubling of variables argument and a careful analysis of the passage to the diagonal. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory. (C) 2018 Elsevier Inc. All rights reserved
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