ON THE CAUCHY PROBLEM FOR BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN HÖLDER SPACES

This paper is concerned with solution in Hölder spaces of the Cauchy problem for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown variables are viewed as deterministic spatial functionals which take values in Banach space...

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Bibliographic Details
Published in:The Annals of probability 2016-01, Vol.44 (1), p.360-398
Main Authors: Tang, Shanjian, Wei, Wenning
Format: Article
Language:English
Subjects:
35R60
60H15
A priori knowledge
backward stochastic differential equations
Backward stochastic partial differential equations
Banach space
Banach spaces
Cauchy problem
Cauchy problems
Differential equations
heat potential
Hölder space
Interpolation
Maximum principle
Partial differential equations
Stochastic models
Stochastic processes
Studies
Uniqueness
Variable coefficients
Vector space
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Summary:This paper is concerned with solution in Hölder spaces of the Cauchy problem for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown variables are viewed as deterministic spatial functionals which take values in Banach spaces of random (vector) processes. We define suitable functional Hölder spaces for them and give some inequalities among these Hölder norms. The existence, uniqueness as well as the regularity of solutions are proved for BSPDEs, which contain new assertions even on deterministic PDEs.
ISSN:0091-1798
2168-894X
DOI:10.1214/14-AOP976