Numerical Schemes for Rough Parabolic Equations

@article{Deya2012NumericalSF,
  title={Numerical Schemes for Rough Parabolic Equations},
  author={Aur'elien Deya},
  journal={Applied Mathematics \& Optimization},
  year={2012},
  volume={65},
  pages={253-292}
}
  • A. Deya
  • Published 2 March 2010
  • Mathematics
  • Applied Mathematics & Optimization
This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0,1) perturbed by a non-linear rough signal. It is the continuation of Deya (Electron. J. Probab. 16:1489–1518, 2011) and Deya et al. (Probab. Theory Relat. Fields, to appear), where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2… 
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References

SHOWING 1-10 OF 49 REFERENCES
Numerical analysis of semilinear stochastic evolution equations in Banach spaces
Partial differential equations driven by rough paths
ANOTHER APPROACH TO SOME ROUGH AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
TLDR
This paper considers general Banach spaces as state spaces and finite dimensional sources of noise, either rough or stochastic, and is able to construct unique solutions of the respective R- and SPDEs by means of a time-dependent transformation of state space and rough path theory.
Stochastic heat equation driven by fractional noise and local time
The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with…
Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II
We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the…
Controlling rough paths
TAYLOR EXPANSIONS OF SOLUTIONS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE
The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not…
Stochastic analysis, rough path analysis and fractional Brownian motions
Abstract. In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the…
Stochastic evolution equations with fractional Brownian motion
Abstract.In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and…
Backward stochastic differential equations with rough drivers
Backward stochastic differential equations (BSDEs) in the sense of Pardoux-Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in…
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