Chilean Probability Seminar “An Invariance Principle for Enhanced Fractional Brownian Motion”

Abstract: Resumen:  Differential equations driven by signals too irregular to be semimartingales arise in many models; fractional Brownian motion with Hurst parameter $H<1/2$ is the classical such driver. For these equations the solution is not a continuous function of the driving path. Rough path theory restores continuity, but only after the driver is enriched with its iterated integrals. A discrete model therefore converges to such an equation only if its enhanced driver converges. Invariance principles at the enhanced level are available, but their limit is Brownian. Enhanced fractional Brownian motion is reached only by construction, either starting from it or imposing a fractional kernel on a random walk. The fractional Donsker theorem, available for over a decade, stops at the level of paths. What is missing is the combination: a fractional limit, reached from a discrete model, at the enhanced level. We supply it. For a centred, strictly stationary, not necessarily Gaussian sequence in $\R^d$ whose normalised partial sums converge finite-dimensionally to fractional Brownian motion with $H\in(1/3,1/2)$, control of covariances and cumulants forces the canonical level-two lift to converge weakly to enhanced fractional Brownian motion in the $\a$-Hölder rough path topology. Continuity of the solution map then transfers the convergence back to the equations we started from.

Join the seminar via Zoom: 
https://reuna.zoom.us/j/84521834914?pwd=OTZ6Y0NWM3pYTGtTbEt3c0luTG96UT09
ID de reunión: 845 2183 4914
Código de acceso: 997973

Speaker: Tomás Laengle (Universidad Humboldt de Berlín).