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Integrability and tail estimates for Gaussian rough differential equations. (English) Zbl 1278.60091
The authors consider the stochastic differential equation \[ dY_t=V(Y_t)dX_t,\;Y_0=y_0 \] driven by a Gaussian process \(X\) and study the derivative – the “Jacobian” – \(DU^X_{t\leftarrow 0}(\cdot)|_{y_0}\) of its flow \( U^X_{t\leftarrow 0}(y_0)=Y_t\). The main result of the paper is the existence of moments of all orders of the Jacobian driven by a certain class of Gaussian processes (including but not restricted to the fractional Brownian motion with Hurst parameter \(H>1/4\)). Moreover, the central estimate the authors derive shows that the logarithm of the Jacobian possesses a tail decaying faster than exponential.
The technical result the authors provide is a central step in order to extend the body of work in rough parts random dynamical equations, as detailed in the paper. Furthermore, the presently developed technique can be applied to more general random dynamical equations, i.e., all such equations that can be controlled by a “greedy” approximation of the local \(p\)-variation. (This concept is detailed in the manuscript.)

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
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