Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients
Copyright policy )Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients
This paper introduces the path derivatives, in the spirit of Dupire's functional It�� calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration and rough PDEs in the same manner as standard stochastic calculus. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients is less regular in the time variable and is not covered in the existing literature. The results are useful for studying viscosity solutions of stochastic PDEs.
36 pages
- University of Michigan–Ann Arbor United States
- University of Michigan–Flint United States
- University of Southern California United States
- University of California System United States
Stochastic integrals, characteristics, Probability (math.PR), path derivatives, Stochastic ordinary differential equations (aspects of stochastic analysis), Itō-Ventzell formula, Mathematics - Analysis of PDEs, Stochastic partial differential equations (aspects of stochastic analysis), functional Itō calculus, 60H05, rough PDEs, Foundations of stochastic processes, FOS: Mathematics, Sample path properties, stochastic PDEs, rough paths, Mathematics - Probability, Analysis of PDEs (math.AP)
Stochastic integrals, characteristics, Probability (math.PR), path derivatives, Stochastic ordinary differential equations (aspects of stochastic analysis), Itō-Ventzell formula, Mathematics - Analysis of PDEs, Stochastic partial differential equations (aspects of stochastic analysis), functional Itō calculus, 60H05, rough PDEs, Foundations of stochastic processes, FOS: Mathematics, Sample path properties, stochastic PDEs, rough paths, Mathematics - Probability, Analysis of PDEs (math.AP)
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