Partial differential equations driven by rough paths

By M. Caruana, Peter K. Friz. 2008

76 citations recorded by the source corpus.

We study a class of linear first and second order partial differential equations driven by weak geometric $p$-rough paths, and prove the existence of a unique solution for these equations. This solution depends continuously on the driving rough path. This allows a robust approach to stochastic partial differential equations. In particular, we may replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all became easy corollaries of the corresponding statements of the driving process. In the case of first order equations with Gaussian noise, we discuss the existence of a density with respect to the Lebesgue measure for the solution.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1dadf17b0234da540c3ae64734bbeb8b2241fcc1

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Citations in the corpus

Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, background): “In the case when both  and  are linear, (Wong—Zakai type) approximations have been studied in great detail [78, 77, 75, 74, 76] and in [20] with rough paths methods.”
Malliavin Calculus for Fractional Delay Equations (2009, methodology): “thanks to rough paths techniques, either by smart transformations allowing to use limiting arguments [3, 4, 8] or by an adaptation of the rough paths formalism to evolution equations [6, 7, 10].”
A priori estimates for rough PDEs with application to rough conservation laws (2016, methodology): “First, Lions, Perthame and Souganidis (see [50,52]) developed a pathwise approach for du + div(A(x, u)) ◦ dW = 0, where W is a continuous real-valued signal and ◦ stands for the Stratonovich product in the Brownian case, then Friz and Gess (see [23]) studied du + div f(t, x, u)dt = F (t, x, u)dt + Λ”
A (rough) pathwise approach to a class of non-linear stochastic partial differential equations (2009, methodology): “As is well known, this ensures that comparison holds for BUC solutions of @t −F � = 0. By applying theorem 1 to this class of linear F we recover the results of [ 5 ].”
Non-linear rough heat equations (2009, background): “As far as this general method is concerned, let us quote the work of Caruana and Friz [5], Caruana et al.”