SRQ seminar – September 27th, 2018

Notes from a talk in the SRQ series.

INI Seminar 20180927. Peter Friz

Rough Paths

References: Lyons '98 (rough path theory born), Gubinelli '04 (controlled paths), F–Victoir '10 [CUP] (geometric rough paths), Gubinelli '10 (branched rough paths), Hairer '14 (regularity structures), Friz–Hairer '14 [Springer] (rough paths + regularity structures)

Example.

$u = u (t)$

$\displaystyle \partial_t u = - u^3 + \xi, \qquad u (0) = 0.$

think about $\xi = \dot{W} \in C^{- 1 / 2 -}$. Picard iteration:

$\displaystyle u^{(1)} = \int_0^{\cdot} \xi =\mathcal{X}^1,$ $\displaystyle u^{(2)} = \int_0^{\cdot} (- X^3 + \xi) = -\mathcal{X}^2 +\mathcal{X}^1,$

$\displaystyle u =\mathcal{X}^1 -\mathcal{X}^2 + \cdots \in C^{1 / 2 -}$

Fun modification. Take $\xi \in \partial_t^3 W_t \in C^{- 5 / 2 -}$ so it is has the same regularity of space–time white noise in three dimensions and take

$\displaystyle \partial_t^2 u = - u^3 + \xi, \qquad u (0) = 0.$

Now the algebra of the above Picard iteration does not change. But power counting gives now

$\displaystyle u =\mathcal{X}^1 -\mathcal{X}^2 + \cdots \in C^{- 1 / 2 -} .$

Another example.

$\displaystyle \mathrm{d} Y = f (Y) \mathrm{d} W \qquad (+ f_0 (Y) \mathrm{d} t)$ $\displaystyle Y^{(1)} = y_0 + \int f (y_0) \mathrm{d} W = y_0 + f (y_0) \mathbb{X}^{\bullet}$ $\displaystyle Y^{(2)} = y_0 + \int f (y_0 + f (y_0) \mathbb{X}^1) \mathrm{d} W = y_0 + f (y_0) X + \int f' (y_0) f (y_0) \mathbb{X}^1 \mathrm{d} W + \cdots$ $\displaystyle = y_0 + f (y_0) X + (f' f) (y_0) \underbrace{\int \mathbb{X}^1 \mathrm{d} W}_{\mathbb{X}^{[\bullet]}} + \cdots$

One more step

$\displaystyle Y^{(3)} = y_0 + \int f (y_0 + f (y_0) \mathbb{X}^{\bullet} + (f' f) (y_0) \mathbb{X}^{[\bullet]} + \cdots) \mathrm{d} W$ $\displaystyle = y_0 + f (y_0) \mathbb{X}^{\bullet} + (f' f) (y_0) \mathbb{X}^{[\bullet]} + \frac{1}{2} f'' (y_0) f (y_0) f (y_0) \underbrace{\int \mathbb{X}^1 \mathbb{X}^1 \mathrm{d} W}_{\mathbb{X}^{[\bullet \bullet]}} + f' (y_0) f' (y_0) f (y_0) \underbrace{\int \mathbb{X}^2 \mathrm{d} W}_{\mathbb{X}^{[[\bullet]]}} + \cdots$

[keywords: Butcher series, branched rough paths]

Note that

$\displaystyle X^{[[\bullet]]} = \int \int \int \mathrm{d} W \mathrm{d} W \mathrm{d} W,$

where integral is over the simplex. On the other hand

$\displaystyle X^{[\bullet \bullet]} = \int \left( \int \mathrm{d} W \right)^2 \mathrm{d} W.$

Multidimensional signals: add indexes everywhere:

$\displaystyle Y^i = y_0^i + f^i_{\alpha} (y_0) \mathbb{X}^{\bullet_{\alpha}} + (f^i_{\alpha ; j} f^j_{\beta}) (y_0) \mathbb{X}^{[\bullet_{\beta}]_{\alpha}} + \frac{1}{2} f^i_{\alpha ; j, k} f^j_{\beta} f^k_{\gamma} (y_0) \mathbb{X}^{[\bullet_{\beta} \bullet_{\gamma}]_{\alpha}} + f^i_{\alpha ; j} f^j_{\beta ; k} f^k_{\gamma} (y_0) \mathbb{X}^{[[\bullet_{\gamma}]_{\beta}]_{\alpha}} + \cdots$

These expansions are very much related to numerical analysis of stochastic differential equations. Truncation at level of $\mathbb{X}^{[\bullet]}$ gives the so-called Milstein scheme.

Theorem 1. Given deterministic objects $\mathbb{W} := (\mathbb{W}^{\bullet}, \mathbb{W}^{[\bullet]}) \in C^{\alpha} \times ?$ with $\alpha \in ?$ where we can think of

$\displaystyle \mathbb{W}^{[\bullet]}_{s, t} = \iint_{s < u < v < t} \mathrm{d} W_v \mathrm{d} W_u$

but which are required only to satisfy the natural algebraic properties of iterated integrals wrt to joining adjacent integration regions (Chen's relation) and that are “analytically good enough”, namely $| \mathbb{W}^{\bullet}_{s, t} | \lesssim | t - s |^{\alpha}$ and $| \mathbb{W}^{[\bullet]}_{s, t} | \lesssim | t - s |^{2 \alpha}$ for some $\alpha \in (1 / 3, 1 / 2)$.

Then, there is a robust theory of the differential equation

$\displaystyle \mathrm{d} Y = f (Y) \mathrm{d} W. $ (1)

This means: there exists a unique solution to (1) driven by $(\mathbb{W}^{\bullet}, \mathbb{W}^{[\bullet]})$ such that there is a good local understanding of such a solution, namely of any small interval $[s, t]$ we have

$\displaystyle Y_t - Y_s = f (Y_s) \mathbb{W}^{\bullet}_{s, t} + (f' f) (Y_s) \mathbb{W}^{[\bullet]}_{s, t} + \underbrace{O (| t - s |^{3 \alpha})}_{\approx o (| t - s |), \text{since $3 \alpha > 1$}} . $

(2)

This already implies that $Y_t$ is a limit of modified Riemann–Stiljes sums.

Actually we can take (2) as a definition for the rough differential equation (1). This approach is due to Davie. Then there is a corresponding existence and uniqueness theorem.

Rough integral.

There exists also a more intrinsic point of view of this equation by introducing a notion of rough integration. We can then look at the differential equation as a fixpoint problem via rough integration. In general we would like to understand an integral of the form

$\displaystyle I (Z) = \int Z \mathrm{d} \mathbb{W}$

where the notation stands to denote that the value of this integral will depend on the full given of the stochastic data $\mathbb{W} := (\mathbb{W}^{\bullet}, \mathbb{W}^{[\bullet]})$. For example we want to integrate $Z = g (W)$. The natural definition looks like modified Riemann sums of the following type

$\displaystyle I (Z) = \int Z \mathrm{d} \mathbb{W} \simeq \sum Z_s \mathbb{W}^{\bullet}_{s, t} + \sum Z_s' \mathbb{W}^{[\bullet]}_{s, t}$

where in the case $Z = g (W)$ we need to take $Z' = g' (W)$ but in general $Z'$ is just a new function for which we are looking abstract conditions which ensure the convergence of the Riemann sums.

Some considerations lead to the condition

$\displaystyle \Delta Z_{s, t} = Z_t - Z_s = Z'_s \mathbb{W}^{\bullet}_{s, t} + \underbrace{R_{s, t}}_{O (| t - s |^{2 \alpha})} .$

This structural assumption (together with the additional requirement $Z' \in C^{\alpha}$) ensure the existence of the limit of Riemann sums. The set of such pairs $(Z, Z')$ is a linear space called space of controlled rough paths. Note that the space of rough paths forms a non-linear (algebraic) manifold. We write $(Z, Z') \in \mathcal{D}_{\mathbb{W}}^{2 \alpha}$ to denote that the space depends on $\mathbb{W}$.

Theorem 2. The rough integral construction of controlled paths as defined right now, works. Namely, as limit of modified Riemann sums, the rough integral exists and comes with a good local description:

$\displaystyle I (Z)_t - I (Z)_s = Z_s \mathbb{W}^{\bullet}_{s, t} + Z'_s \mathbb{W}^{[\bullet]}_{s, t} + O (| t - s |^{3 \alpha}) .$

The integral is continuous wrt. $(Z, Z')$ and $\mathbb{W}$.

Remark 3. Taking the distributional derivative of $\int \operatorname{Zd}\mathbb{W}$ we effectively constructed the product $Z \dot{W}$, which classicaly makes no sense since the product is not well defined on $C^{\alpha} \times C^{\alpha - 1}$ when $\alpha < 1 / 2$. All we are saying is that, provided we have a meaningful definition of $W \dot{W}$ we can extend it to build the product $Z \dot{W}$ for all controlled paths $(Z, Z') \in \mathcal{D}_{\mathbb{W}}^{2 \alpha}$.

Link with probability.

At this stage there is no probability involved in these considerations. This construction is consistent with Ito or Stratonovich integrations. Namely, if I'm in a setting where both integrals (rough/stochastic) exists I can make them agree.

Precisely. We need for $W$ to be a Brownian motion (or a semi–martingale) and $Z$ to be adapted (note: anticipating stoch calculus is trivial with rough paths).

Examples. Take $i = 1, 2$,

$\displaystyle \hat{W}_t^i = \int_0^t | t - s |^{H - 1 / 2} \mathrm{d} W_s^i$

with $H \in (0, 1 / 2]$. Then $\hat{W}^i \in C^{H -}$. Consider two IID copies $\hat{W} = (\hat{W}^1, \hat{W}^2)$ (2d fractional Brownian motion like process). We can constuct a rough path $\hat{\mathbb{W}}$ over $\hat{W}$ as long as $H > 1 / 4$. Take moreover $H > 1 / 3$ so the above theory applies. Now $\int f (\hat{W}) \mathrm{d} \hat{\mathbb{W}}$ can be constructed as a rough integral.

However if I consider $\int f (\hat{W}) \mathrm{d} W$ there is no way of constructing it as a rough integral over $\mathbb{W}$ when $H < 1 / 2$. The right thing to do is to be able to define $\int \hat{W} \mathrm{d} W$ and a full rough path over the pair $(W, \hat{W})$ which will contain also the mixed product $\int \hat{W} \mathrm{d} W$. But if the regularity is bad then we will need also

$\displaystyle \int (\hat{W})^2 \mathrm{d} W, \int (\hat{W})^3 \mathrm{d} W, \cdots$

and so on. In this very simple example regularization of the integral $\int f (\hat{W}) \mathrm{d} W$ will not converge, in particular

$\displaystyle \int \hat{W}^{\varepsilon} \mathrm{d} W^{\varepsilon}$

will not converge and has to be renormalized by removing a diverging correction when $H < 1 / 2$. Namely the Ito–Stratonovich correction in this setting is infinite.

Tweaking the second level of a rough path.

$\mathbb{W} := (\mathbb{W}^{\bullet}, \mathbb{W}^{[\bullet]})$. Take $\tilde{\mathbb{W}} := (\tilde{\mathbb{W}}^{\bullet}, \tilde{\mathbb{W}}^{[\bullet]})$ where $\tilde{\mathbb{W}}^{\bullet} =\mathbb{W}^{\bullet}$ and $\tilde{\mathbb{W}}^{[\bullet]} =\mathbb{W}^{[\bullet]} + \Gamma$ where $\Gamma \in C^1$. Then if we solve

$\displaystyle \mathrm{d} Y = F (Y) \mathrm{d} \tilde{\mathbb{W}} + f_0 (Y) \mathrm{d} t$

we have that $Y$ is also a solution to

$\displaystyle \mathrm{d} Y = F (Y) \mathrm{d} \mathbb{W}+ f_0 (Y) \mathrm{d} t + F' F (Y) \mathrm{d} \Gamma .$ $\displaystyle \begin{array}{lccc} & \{ f_0, f \} \times \mathbb{W} & \xrightarrow[\operatorname{RDE}]{} & (Y, f (Y)) \in \operatorname{cRP}\\ & {\mbox{\rotatebox[origin=c]{90}{$\longrightarrow$}}}_{\operatorname{lift}} & & {\mbox{\rotatebox[origin=c]{-90}{$\longrightarrow$}}}\\ & \{ f_0, f \} \times W & \xrightarrow[\operatorname{ODE}]{} & Y \end{array}$

Changing the lift $\mathbb{W}$ into $\tilde{\mathbb{W}}$ can be reabsorbed into a change in the data:

$\displaystyle \begin{array}{lccc} & \{ f_0, f \} \times {\tilde{\mathbb{W}}} & \xrightarrow[\operatorname{RDE}]{} & (Y, f (Y)) \in \operatorname{cRP}\\ & {\mbox{\rotatebox[origin=c]{90}{$\longrightarrow$}}}_{\operatorname{lift}} & & \\ & {\{ f_0 + f' f, f \}} \times W & \xrightarrow[\operatorname{ODE}]{} & Y \end{array}$

Relation with regularity structures

$\displaystyle \begin{array}{lcl} \textbf{Rough Paths} & & \textbf{Regularity structure}\\ & & \\ \text{Chen} & \longleftrightarrow & \text{Structure group}\\ \text{RP} & \longleftrightarrow & \text{model}\\ \text{cRP} & \longleftrightarrow & \text{modelled distributions}\\ \text{rough integral} & \longleftrightarrow & \text{reconstruction}\\ & \text{continuity} & \\ \text{RP higher order translation} & \longleftrightarrow & \text{renormalization} \end{array}$

Renormalization of RP is treated in the recent paper: Bruned–Chevyrev–Friz–Preiss.

$\displaystyle $