Abstract
Partial differential equations driven by rough paths are studied. We return to the investigations of [Caruana, Friz and Oberhauser: A (rough) pathwise approach to a class of non- linear SPDEs, Annales de l’Institut Henri Poincaré/Analyse Non Linéaire 2011, 28, pp. 27–46], motivated by the Lions–Souganidis theory of viscosity solutions for SPDEs. We continue and complement the previous (uniqueness) results with general existence and regularity statements. Much of this is transformed to questions for deterministic parabolic partial differential equations in viscosity sense. On a technical level, we establish a refined parabolic theorem of sums which may be useful in its own right.
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Notes
- 1.
Using any of the equivalent norms on \(S^n\).
- 2.
If one assumes that given \(u\in \mathrm {BUSC}\left( [0,T]\times \mathbb {R} ^{n}\right) ,\) \(v\in \mathrm {BLSC}\left( [0,T]\times \mathbb {R}^{n}\right) \) satisfy (6) on \((0,T]\times \mathbb {R}^{n}\) then it already follows that (5) holds true. This follows from the so-called Accessibility Theorem [7].
- 3.
As is well-known, the precise meaning of (6) is expressed (equivalently) in terms of “touching” test-functions or in term of sub- and super-jets. We shall switch between these points of view without further comments.
- 4.
All results of Lions–Souganidis on stochastic viscosity theory are stated in \(\mathrm {BUC}\)-spaces with spatial domain \(\mathbb {R}^n\).
- 5.
These ideas may prove useful in establishing rates of convergence for equations driven by \((z^{\epsilon })\), convergent in rough path sense.
- 6.
The point here, of course, is to handle appropriately terminal time \(T\) which is a well-documented subtlety in (parabolic) viscosity theory; some mistakes in the early literature were corrected in Ref. [7].
- 7.
... notably boundedness of \(F\left( \cdot ,\cdot ,y,p,X\right) \) when \(y,p,X\) remain in a bounded set ...
- 8.
In the strict terminology of rough path theory: geometric \(\alpha \)-Hölder rough paths.
- 9.
... in which case (47) is understood in Stratonovich form.
- 10.
This may be expressed in terms of \(F\); in particular \(F\) then satisfies \( \Phi ^{\left( 3\right) }\)-invariant comparison as introduced in Ref. [6]; there it was also checked that these structural assumptions are satisfied by many examples ; notably those arising from pathwise stochastic control theory.
- 11.
Consistency of \(L^{2}\)-theory with RPDE theory has been established in Ref. [16].
- 12.
Above regularity assumptions can be weakened with the “joint lift” approach carried out in Ref. [13].
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Acknowledgments
J. Diehl is supported by DFG Priority Program 1324; P. Friz and H. Oberhauser received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement nr. 258237. P. Friz would like to thank G. Barles for a helpful email exchange, E. Jakobsen for pointing us to [20, 21], M. Soner and T. Souganidis for discussions during the Oberwolfach meetings on Stochastic Analysis in Finance and Insurance and Rough Paths and PDEs.
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Diehl, J., Friz, P.K., Oberhauser, H. (2014). Regularity Theory for Rough Partial Differential Equations and Parabolic Comparison Revisited. In: Crisan, D., Hambly, B., Zariphopoulou, T. (eds) Stochastic Analysis and Applications 2014. Springer Proceedings in Mathematics & Statistics, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-319-11292-3_8
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