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Convergence rates for the full gaussian rough paths

Peter Friz, Sebastian Riedel (2014)

Annales de l'I.H.P. Probabilités et statistiques

Under the key assumption of finite ρ -variation, ρ ∈ [ 1 , 2 ) , of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), ρ = 1 resp. ρ = 1 / ( 2 H ) , we recover and extend the respective results of (Trans. Amer. Math. Soc.361 (2009) 2689–2718) and (Ann. Inst. Henri Poincasé Probab. Stat.48(2012) 518–550). In particular, we establish an a.s. rate k - ( 1 / ρ - 1 / 2 - ε ) , any ε g t ; 0 , for Wong–Zakai and Milstein-type...

Convex hulls, Sticky particle dynamics and Pressure-less gas system

Octave Moutsinga (2008)

Annales mathématiques Blaise Pascal

We introduce a new condition which extends the definition of sticky particle dynamics to the case of discontinuous initial velocities u 0 with negative jumps. We show the existence of a stochastic process and a forward flow φ satisfying X s + t = φ ( X s , t , P s , u s ) and d X t = E [ u 0 ( X 0 ) / X t ] d t , where P s = P X s - 1 is the law of X s and u s ( x ) = E [ u 0 ( X 0 ) / X s = x ] is the velocity of particle x at time s ≥ 0 . Results on the flow characterization and Lipschitz continuity are also given.Moreover, the map ( x , t ) ↦ M ( x , t ) : = P ( X t ≤ x ) is the entropy solution of a scalar conservation law ∂ t M + ∂ x ( A ( M ) ) = 0 where the flux A represents the particles...

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