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- ItemEikonal equations and pathwise solutions to fully non-linear SPDEs(New York, NY : Springer, 2016) Friz, Peter K.; Gassiat, Paul; Lions, Pierre-Louis; Souganidis, Panagiotis E.We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors.
- ItemSingular paths spaces and applications(Philadelphia, Pa. : Taylor & Francis, 2021) Bellingeri, Carlo; Friz, Peter K.; Gerencsér, MátéMotivated by recent applications in rough volatility and regularity structures, notably the notion of singular modeled distribution, we study paths, rough paths and related objects with a quantified singularity at zero. In a pure path setting, this allows us to leverage on existing SLE Besov estimates to see that SLE traces takes values in a singular Hölder space, which quantifies a well-known boundary effect in the regime κ<1. We then consider the integration theory against singular rough paths and some extensions thereof. This gives a method to reconcile, from a regularity structure point of view, different singular kernels used to construct (fractional) rough volatility models and an effective reduction to the stationary case which is crucial to apply general renormalization methods.