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Semigroup solution of path-dependent second-order parabolic partial differential equations. (English) Zbl 1412.60091

Summary: We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K10 Second-order parabolic equations
60H07 Stochastic calculus of variations and the Malliavin calculus
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