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Pathwise Taylor expansions for Itô random fields

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  • In this paper we study the pathwise stochastic Taylor expansion, in the sense of our previous work [3], for a class of Itô-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work [3], our new expansion can be defined around any random time-space point (τ,ξ), where the temporal component τ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point (τ,ξ). As an application, we show how this new form of pathwise Taylor expansion could lead to a different treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence lead to a definition of the stochastic viscosity solution for such SPDEs, which is new in the literature, and potentially of essential importance in stochastic control theory.
    Mathematics Subject Classification: Primary: 60H07, 15, 30; Secondary: 35R60, 34F05.


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