Fractional Brownian Motion, Non-Semimartingales, and the Lift¶

Job: develop the fBM (Hurst $H$, covariance, $p$-variation), the canonical rough path lift ($H>1/4$), the non-semimartingale, and the RDE driven by fBM — with the fBM $H>1/4$ canonical lift / RDE as the worked example.

Core¶

The fractional Brownian motion (fBM)¶

fBM / $H$ / $p$-variation / rough	Description
fBM $B^H$	the Hurst $H$, the covariance $r(t,s)=\frac12(
$p$-variation	$p>1/H$ (the $H$ the Hurst, the $p$ the roughness)
Rough path	$H>1/4$ the canonical rough path (the $H>1/3$ the level-2)

The lift and the canonical¶

Lift / canonical / $H>1/4$ / level-2	Description
Canonical lift	$H>1/4$ the canonical rough path lift (the $\mathbb{B}^H$ the iterated)
Level-2	$H>1/3$ the level-2 suffices (the $H\in(1/4,1/3]$ the level-3)
Itô / Stratonovich	the BM $H=1/2$, the Itô / Stratonovich the $\mathbb{B}$ (the stochastic processes)

The non-semimartingale and the rough path¶

Non-semimartingale / fBM / rough	Description
Non-semimartingale	$H\ne1/2$ the fBM not the semimartingale (the stochastic processes)
Rough path	the fBM the rough path (the $H>1/4$ the canonical, the non-semimartingale)

The RDE driven by fBM¶

RDE fBM / $H>1/4$ / Gaussian / SPDE	Description
RDE	$dY=f(Y)\,dB^H$ the $H>1/4$ the well-posed (the SPDE)
Gaussian	the Gaussian the rough (the $H$ the Hurst, the Gaussian the driver)

See it¶

fBM $B^H$ for $H>1/4$ the canonical rough path and the RDE. The fBM $B^H$ the Hurst $H$, the covariance, the $p$-variation $p>1/H$. The canonical lift: the $H>1/4$ the canonical rough path (the $\mathbb{B}^H$ the iterated, the $H>1/3$ the level-2). The RDE: the $dY=f(Y)\,dB^H$ the $H>1/4$ the well-posed (the SPDE). The key point: the fBM the canonical rough path ($H>1/4$), the RDE the well-posed — the non-semimartingale the rough path.

Check¶

Show that the fBM $B^H$ for $H>1/4$ admits a canonical rough path, and explain the non-semimartingale.

The fBM $B^H$: the Hurst $H$, the covariance $r(t,s)=\frac12(|t|^{2H}+|s|^{2H}-|t-s|^{2H})$, the $p$-variation $p>1/H$. The canonical lift: the $H>1/4$ the $\mathbb{B}^H$ the iterated integral (the Gaussian, the $H>1/4$ the convergence, the $H>1/3$ the level-2 the $H\in(1/4,1/3]$ the level-3). The non-semimartingale: the $H\ne1/2$ the fBM not the semimartingale (the stochastic processes, the $H<1/2$ the not-martingale, the $H>1/2$ the not-semimartingale). So the fBM the canonical ($H>1/4$), the non-semimartingale.

Explain the RDE driven by fBM and the well-posedness.

The RDE: $dY=f(Y)\,dB^H$ (the fBM the driver, the $f\in C^{[p]+1}$). The well-posedness: the $H>1/4$ the canonical rough path $\mathbf{B}^H$ (the $p>1/H<4$ the rough path), the Itô–Lyons the $\mathbf{B}^H\mapsto Y$ the continuous (the controlled-and-sewing), the existence / uniqueness. The $H\le1/4$ the open (the canonical the not-converge, the SPDE the renormalization). So the fBM the RDE ($H>1/4$), the well-posed.

Pitfalls¶

The fBM the $H$ (the Hurst, the covariance, the $p>1/H$) — the $H$ the roughness, the $p$ the variation.
The canonical $H>1/4$ (the $\mathbb{B}^H$ the iterated, the $H>1/3$ the level-2) — the $H>1/4$ the canonical, the $H\le1/4$ the open.
The non-semimartingale ($H\ne1/2$, the stochastic processes) — the fBM the not-semimartingale, the rough path the pathwise.
The RDE the $H>1/4$ (the well-posed, the Itô–Lyons) — the $H>1/4$ the well-posed, the $H\le1/4$ the open / the SPDE.

Refs¶

Mandelbrot–Van Ness, "Fractional Brownian motions, fractional noises" (SIAM 1968); the fBM
Coutin–Qian, "Stochastic differential equations driven by fBM" (2002); the $H>1/4$ canonical
Friz–Victoir, Multidimensional Stochastic Processes as Rough Paths (2010); the fBM the lift

Next¶

RP index · Foundations · Signature & Features · Controlled & Sewing · Branched & Hopf · Stochastic processes · SPDE

fBM / \(H\) / \(p\)-variation / rough	Description
fBM \(B^H\)	the Hurst \(H\), the covariance $r(t,s)=\frac12(
\(p\)-variation	\(p>1/H\) (the \(H\) the Hurst, the \(p\) the roughness)
Rough path	\(H>1/4\) the canonical rough path (the \(H>1/3\) the level-2)