On the Kolmogorov equation associated with Volterra equations and Fractional Brownian Motion

We consider a Volterra convolution equation in $\mathbb{R}^d$ perturbed with an additive fractional Brownian motion of Riemann-Liouville type with Hurst parameter $H\in (0,1)$. We show that its solution solves a stochastic partial differential equation (SPDE) in the Hilbert space of square-integrable functions. Such an equation motivates our study of an unconventional class of SPDEs requiring an original extension of the drift operator and its Fr\'echet differentials. We prove that these SPDEs generate a Markov stochastic flow which is twice Fr\'echet differentiable with respect to the initial data. This stochastic flow is then employed to solve, in the classical sense of infinite dimensional calculus, the path-dependent Kolmogorov equation corresponding to the SPDEs. In particular, we associate a time-dependent infinitesimal generator with the fractional Brownian motion. In the final section, we show some obstructions in the analysis of the mild formulation of the Kolmogorov equation for SPDEs driven by the same infinite dimensional noise. This problem, which is relevant to the theory of regularization-by-noise, remains open for future research

Statistical analysis of the mixed fractional Ornstein--Uhlenbeck process

This paper addresses the problem of estimating drift parameter of the Ornstein - Uhlenbeck type process, driven by the sum of independent standard and fractional Brownian motions. The maximum likelihood estimator is shown to be consistent and asymptotically normal in the large-sample limit, using some recent results on the canonical representation and spectral structure of mixed processes.Comment: to appear in Theory of Probability and its Application

A Pathwise Fractional one Compartment Intra-Veinous Bolus Model

Extending deterministic compartments pharmacokinetic models as diffusions seems not realistic on biological side because paths of these stochastic processes are not smooth enough. In order to extend one compartment intra-veinous bolus models, this paper suggests to modelize the concentration process $C$ by a class of stochastic differential equations driven by a fractional Brownian motion of Hurst parameter belonging to $]1/2,1[$. The first part of the paper provides probabilistic and statistical results on the concentration process $C$ : the distribution of $C$, a control of the uniform distance between $C$ and the solution of the associated ordinary differential equation, an ergodic theorem for the concentration process and its application to the estimation of the elimination constant, and consistent estimators of the driving signal's Hurst parameter and of the volatility constant. The second part of the paper provides applications of these theoretical results on simulated concentration datas : a qualitative procedure for choosing parameters on small sets of observations, and simulations of the estimators of the elimination constant and of the driving signal's Hurst parameter. The relationship between the estimations quality and the size/length of the sample is discussed.Comment: 16 pages, 6 figure