Lecture 10. Estimating iterated integrals (Part 1)

In the previous lecture we introduced the signature of a bounded variation path x as the formal series
\mathfrak{S} (x)_{s,t} =1 + \sum_{k=1}^{+\infty} \int_{\Delta^k [s,t]}  dx^{\otimes k}.
If now x \in C^{p-var}([0,T],\mathbb{R}^d), p \ge 1 the iterated integrals \int_{\Delta^k [s,t]}  dx^{\otimes k} can only be defined as Young integrals when p < 2. In this lecture, we are going to derive some estimates that allow to define the signature of some (not all) paths with a finite p variation when p \ge 2. These estimates are due to Terry Lyons in his seminal paper and this is where the rough paths theory really begins.

For P \in \mathbb{R} [[X_1,...,X_d]] that can be writen as
P=P_0+\sum_{k = 1}^{+\infty} \sum_{I \in \{1,...,d\}^k}a_{i_1,...,i_k} X_{i_1}...X_{i_k},
we define
\| P \| =|P_0|+\sum_{k = 1}^{+\infty} \sum_{I \in \{1,...,d\}^k}|a_{i_1,...,i_k}|  \in [0,\infty].
It is quite easy to check that for P,Q \in \mathbb{R} [[X_1,...,X_d]]
\| PQ \| \le \| P \| \| Q\|.
Let x \in C^{1-var}([0,T],\mathbb{R}^d). For p \ge 1, we denote
\left\|  \int dx^{\otimes k}\right\|_{p-var, [s,t]}=\left( \sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{i=0}^{n-1} \left\|  \int_{\Delta^k [t_i,t_{i+1}]}  dx^{\otimes k} \right\|^p \right)^{1/p},
where \mathcal{D}[s,t] is the set of subdivisions of the interval [s,t]. Observe that for k \ge 2, in general
\int_{\Delta^k [s,t]}  dx^{\otimes k}+ \int_{\Delta^k [t,u]}  dx^{\otimes k} \neq \int_{\Delta^k [s,u]}  dx^{\otimes k}.
Actually from the Chen’s relations we have
\int_{\Delta^n [s,u]}  dx^{\otimes n}= \int_{\Delta^n [s,t]}  dx^{\otimes k}+ \int_{\Delta^n [t,u]}  dx^{\otimes k} +\sum_{k=1}^{n-1} \int_{\Delta^k [s,t]}  dx^{\otimes k }\int_{\Delta^{n-k} [t,u]}  dx^{\otimes (n-k) }.
It follows that \left\|  \int dx^{\otimes k}\right\|_{p-var, [s,t]} needs not to be the p-variation of t \to \int_{\Delta^k [s,t]} dx^{\otimes k}.
The first major result of rough paths theory is the following estimate:

Proposition: Let p \ge 1. There exists a constant C \ge 0, depending only on p, such that for every x \in C^{1-var}([0,T],\mathbb{R}^d) and k  \ge 0,
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k} \right\| \le \frac{C^k}{\left( \frac{k}{p}\right)!} \left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]}  \right)^k, \quad 0 \le s \le t \le T.

By \left( \frac{k}{p}\right)!, we of course mean \Gamma  \left( \frac{k}{p}+1\right). Some remarks are in order before we prove the result. If p=1, then the estimate becomes
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k} \right\| \le \frac{C^k}{k!} \| x \|_{1-var, [s,t]}^k,
which is immediately checked because
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k} \right\|
\le \sum_{I \in \{1,...,d\}^k} \left\|   \int_{\Delta^{k}[s,t]}dx^{I} \right\|
\le \sum_{I \in \{1,...,d\}^k}  \int_{s \le t_1 \le t_2 \le \cdots \le t_k \le t} \| dx^{i_1}(t_1) \| \cdots \| dx^{i_k}(t_k)\|
\le \frac{1}{k!} \left( \sum_{j=1}^ d \| x^j \|_{1-var, [s,t]} \right)^k.

We can also observe that for k \le p, the estimate is easy to obtain because
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k} \right\| \le \left\|  \int dx^{\otimes k}\right\|_{\frac{p}{k}-var, [s,t]}.
So, all the work is to prove the estimate when k >p. The proof is split into two lemmas. The first one is a binomial inequality which is actually quite difficult to prove:

Lemma: For x,y >0, n \in \mathbb{N}, n \ge 0, and p \ge 1,
\sum_{j=0}^n \frac{x^{j/p}}{\left( \frac{j}{p}\right)!} \frac{y^{(n-j)/p}}{\left( \frac{n-j}{p}\right)!} \le p \frac{(x+y)^{n/p}}{ {\left( \frac{n}{p}\right)!}}.

Proof: See Lemma 2.2.2 in the article by Lyons or this proof for the sharp constant \square

The second one is a lemma that actually already was essentially proved in the Lecture on Young’s integral, but which was not explicitly stated.

Lemma: Let \Gamma: \{ 0 \le s \le t \le T \} \to \mathbb{R}^N. Let us assume that:

  • There exists a control \tilde{\omega} such that
    \lim_{r \to 0} \sup_{(s,t), \tilde{\omega}(s,t) \le r } \frac{\| \Gamma_{s,t} \|}{r}=0;
  • There exists a control \omega and \theta >1, \xi >0 such that for 0 \le s \le t \le u \le T,
    \| \Gamma_{s,u} \| \le \| \Gamma_{s,t} \|+ \| \Gamma_{t,u} \| +\xi \omega(s,u)^\theta.

Then, for all 0 \le s \le t \le T,
\| \Gamma_{s,t} \| \le \frac{\xi}{1-2^{1-\theta}} \omega(s,t)^\theta.

Proof:
See the proof of the Young-Loeve estimate or Lemma 6.2 in the book by Friz-Victoir \square

We can now turn to the proof of the main result.

Proof:
Let us denote
\omega(s,t)=\left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]}  \right)^p.
We claim that \omega is a control. Indeed for 0 \le s \le t \le u \le T, we have from Holder’s inequality
\omega(s,t)+\omega(t,u)
= \left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]}  \right)^p+\left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [t,u]}  \right)^p
\le \left( \sum_{j=1}^{[p]}\left(  \left\|  \int dx^{\otimes j}\right\|^{p/j}_{\frac{p}{j}-var, [s,t]} +   \left\|  \int dx^{\otimes j}\right\|^{p/j}_{\frac{p}{j}-var, [t,u]}\right)^{1/p} \right)^p
\le \left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,u]}  \right)^p =\omega(s,u).

It is clear that for some constant \beta > 0 which is small enough, we have for k \le p,
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k} \right\|  \le \frac{1}{\beta \left( \frac{k}{p}\right)!} \omega(s,t)^{k/p}.

Let us now consider
\Gamma_{s,t}=  \int_{\Delta^{[p]+1} [s,t]}  dx^{\otimes ([p]+1)}.
From the Chen’s relations, for 0 \le s \le t \le u \le T,
\Gamma_{s,u}=  \Gamma_{s,t}+  \Gamma_{t,u}+\sum_{j=1}^{[p]} \int_{\Delta^j [s,t]}  dx^{\otimes j }\int_{\Delta^{[p]+1-j} [t,u]}  dx^{\otimes ([p]+1-j) }.
Therefore,
\| \Gamma_{s,u}\|
\le \|  \Gamma_{s,t} \| + \|  \Gamma_{t,u} \| +\sum_{j=1}^{[p]} \left\| \int_{\Delta^j [s,t]}  dx^{\otimes j }\right\|  \left\| \int_{\Delta^{[p]+1-j} [t,u]}  dx^{\otimes ([p]+1-j) }\right\|
\le  \|  \Gamma_{s,t} \| + \|  \Gamma_{t,u} \| +\frac{1}{\beta^2}  \sum_{j=1}^{[p]} \frac{1}{ \left( \frac{j}{p}\right)!} \omega(s,t)^{j/p}\frac{1}{ \left( \frac{[p]+1-j}{p}\right)!} \omega(t,u)^{([p]+1-j)/p}
\le \|  \Gamma_{s,t} \| + \|  \Gamma_{t,u} \| +\frac{1}{\beta^2}  \sum_{j=0}^{[p]+1} \frac{1}{ \left( \frac{j}{p}\right)!} \omega(s,t)^{j/p}\frac{1}{ \left( \frac{[p]+1-j}{p}\right)!} \omega(t,u)^{([p]+1-j)/p}
\le \|  \Gamma_{s,t} \| + \|  \Gamma_{t,u} \| +\frac{1}{\beta^2} p  \frac{(\omega(s,t)+\omega(t,u))^{([p]+1)/p}}{ {\left( \frac{[p]+1}{p}\right)!}}
\le  \|  \Gamma_{s,t} \| + \|  \Gamma_{t,u} \| +\frac{1}{\beta^2} p  \frac{\omega(s,u)^{([p]+1)/p}}{ {\left( \frac{[p]+1}{p}\right)!}}.
On the other hand, we have
\|  \Gamma_{s,t} \| \le A \| x \|_{1-var,[s,t]}^{[p]+1}.
We deduce from the previous lemma that
\|  \Gamma_{s,t} \| \le \frac{1}{\beta^2} \frac{p}{1-2^{1-\theta}}  \frac{\omega(s,t)^{([p]+1)/p}}{ {\left( \frac{[p]+1}{p}\right)!}},
with \theta=\frac{[p]+1}{p}. The general case k \ge p is dealt by induction. The details are let to the reader \square

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4 Responses to Lecture 10. Estimating iterated integrals (Part 1)

  1. Taras says:

    There is a misprint in the definition of the p-variation of iterated integral.
    Could you please clarify, how you use Holder inequality in the part of proof the main theorem, that corresponds to the statement that smth is control? I believe, you want to prove that (\sum a_j)^p+(\sum b_j)^p\leq (\sum (a_j^p+b^p_j)^{1/p})^p, however I do not understand how it follows from Holder inequality. Thanks!

  2. Michael Hartig says:

    First of all, thanks a lot for this lecture, it’s helping me quite a bit.
    Can you please verify, that in the definition of the p-variation the running index of the sum on the right hand side should be i instead of k?
    \left\| \int dx^{\otimes k}\right\|_{p-var, [s,t]}=\left( \sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{k=0}^{n-1} \left\| \int_{\Delta^k [t_i,t_{i+1}]} dx^{\otimes k} \right\|^p \right)^{1/p},

  3. Thanks. This is now corrected.

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