Lecture 11. Estimating iterated integrals (Part 2)

Let x \in C^{1-var}([0,T],\mathbb{R}^d). Since
\omega(s,t)=\left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]}  \right)^p
is a control, the estimate
\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.
easily implies that for k > p,
\left\|  \int  dx^{\otimes k} \right\|_{1-var, [s,t]}   \le \frac{C^k}{\left( \frac{k}{p}\right)!} \omega(s,t)^{k/p}.
We stress that it does not imply a bound on the 1-variation of the path t \to   \int_{\Delta^k [0,t]}  dx^{\otimes k} . What we can get for this path, are bounds in p-variation:

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 [0,\cdot]}  dx^{\otimes k} \right\|_{p-var, [s,t]}  \le \frac{C^k}{\left( \frac{k}{p}\right)!} \omega(s,t)^{1/p} \omega(0,T)^{\frac{k-1}{p}}
where
\omega(s,t)= \left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]}  \right)^p, \quad 0 \le s \le t \le T.

Proof: This is an easy consequence of the Chen’s relations. Indeed,

\left\| \int_{\Delta^k [0,t]}  dx^{\otimes k} - \int_{\Delta^k [0,s]}  dx^{\otimes k} \right\|
=\left\| \sum_{j=1}^k  \int_{\Delta^j [s,t]}  dx^{\otimes j} \int_{\Delta^{j-k} [0,s]}  dx^{\otimes (k-j)} \right\|
\le  \sum_{j=1}^k \left\|  \int_{\Delta^j [s,t]}  dx^{\otimes j}  \right\| \left\|  \int_{\Delta^{j-k} [0,s]}  dx^{\otimes (k-j)} \right\|
\le C^k \sum_{j=1}^k  \frac{1}{\left( \frac{j}{p}\right)!} \omega(s,t)^{j/p}  \frac{1}{\left( \frac{k-j}{p}\right)!} \omega(s,t)^{(k-j)/p}
\le C^k \omega(s,t)^{1/p} \sum_{j=1}^k  \frac{1}{\left( \frac{j}{p}\right)!} \omega(0,T)^{(j-1)/p}  \frac{1}{\left( \frac{k-j}{p}\right)!} \omega(0,T)^{(k-j)/p}
\le  C^k \omega(s,t)^{1/p} \omega(0,T)^{(k-1)/p}\sum_{j=1}^k  \frac{1}{\left( \frac{j}{p}\right)!}   \frac{1}{\left( \frac{k-j}{p}\right)!}.
and we conclude with the binomial inequality \square

We are now ready for a second major estimate which is the key to define iterated integrals of a path with p-bounded variation when p \ge 2.

Theorem: Let p \ge 1, K > 0 and x,y \in C^{1-var}([0,T],\mathbb{R}^d) such that
\sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}-   \int  dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}  \le 1,
and
\left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]}  \right)^p+ \left( \sum_{j=1}^{[p]} \left\|  \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]}  \right)^p \le K.
Then there exists a constant C \ge 0 depending only on p and K such that for 0\le s \le t \le T and k \ge 1
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k}-  \int_{\Delta^k [s,t]}  dy^{\otimes k} \right\|  \le \left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}-   \int  dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}  \right) \frac{C^k}{\left( \frac{k}{p}\right)!} \omega(s,t)^{k/p} ,
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k}\right\| +\left\|  \int_{\Delta^k [s,t]}  dy^{\otimes k} \right\|  \le  \frac{C^k}{\left( \frac{k}{p}\right)!} \omega(s,t)^{k/p}
where \omega is the control
\omega(s,t)=  \frac{ \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 dy^{\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, [0,T]}  \right)^p+ \left( \sum_{j=1}^{[p]} \left\|  \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]}  \right)^p }
+\left( \frac{\sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j} -  \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} }{\sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j} -  \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]} }  \right)^p

Proof: We prove by induction on k that for some constants C,\beta,
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k}-  \int_{\Delta^k [s,t]}  dy^{\otimes k} \right\|  \le \left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}-   \int  dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}  \right) \frac{C^k}{\beta \left( \frac{k}{p}\right)!} \omega(s,t)^{k/p},
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k}\right\| +\left\|  \int_{\Delta^k [s,t]}  dy^{\otimes k} \right\|  \le  \frac{C^k}{\beta \left( \frac{k}{p}\right)!} \omega(s,t)^{k/p}

For k \le p, we trivially have
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k}-  \int_{\Delta^k [s,t]}  dy^{\otimes k} \right\| \le  \left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}-   \int  dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}  \right)^k \omega(s,t)^{k/p}
\le   \left( \sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}-   \int  dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}  \right) \omega(s,t)^{k/p}.
and
\left\|  \int_{\Delta^k [s,t]}  dx^{\otimes k}\right\| +\left\|  \int_{\Delta^k [s,t]}  dy^{\otimes k} \right\|  \le  K^{k/p} \omega(s,t)^{k/p}.
Not let us assume that the result is true for 0 \le j \le k with k > p. Let
\Gamma_{s,t}=\int_{\Delta^k [s,t]}  dx^{\otimes (k+1)}-  \int_{\Delta^k [s,t]}  dy^{\otimes (k+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}^{k} \int_{\Delta^j [s,t]}  dx^{\otimes j }\int_{\Delta^{k+1-j} [t,u]}  dx^{\otimes (k+1-j) }-\sum_{j=1}^{k} \int_{\Delta^j [s,t]}  dy^{\otimes j }\int_{\Delta^{k+1-j} [t,u]}  dy^{\otimes (k+1-j) }.
Therefore, from the binomial inequality
\| \Gamma_{s,u}\|
\le   \|  \Gamma_{s,t} \| + \|  \Gamma_{t,u} \| +\sum_{j=1}^{k} \left\| \int_{\Delta^j [s,t]}  dx^{\otimes j }- \int_{\Delta^j [s,t]}  dy^{\otimes j } \right\|  \left\| \int_{\Delta^{k+1-j} [t,u]}  dx^{\otimes (k+1-j) }\right\|
+\sum_{j=1}^{k} \left\| \int_{\Delta^{j} [s,t]}  dy^{\otimes j }\right\|   \left\| \int_{\Delta^{k+1-j} [t,u]}  dx^{\otimes (k+1-j) }-   \int_{\Delta^{k+1-j} [t,u]}  dy^{\otimes (k+1-j) } \right\|
\le   \|  \Gamma_{s,t} \| + \|  \Gamma_{t,u} \| +\frac{1}{\beta^2}\tilde{\omega}(0,T) \sum_{j=1}^{k}   \frac{C^j}{\left( \frac{j}{p}\right)!} \omega(s,t)^{j/p}   \frac{C^{k+1-j}}{\left( \frac{k+1-j}{p}\right)!} \omega(t,u)^{(k+1-j)/p}
+\frac{1}{\beta^2}\tilde{\omega}(0,T) \sum_{j=1}^{k}   \frac{C^j}{\left( \frac{j}{p}\right)!} \omega(s,t)^{j/p}   \frac{C^{k+1-j}}{\left( \frac{k+1-j}{p}\right)!} \omega(t,u)^{(k+1-j)/p}
\le   \|  \Gamma_{s,t} \| + \|  \Gamma_{t,u} \| +\frac{2p}{\beta^2} \tilde{\omega}(0,T) C^{k+1} \frac{ \omega(s,u)^{(k+1)/p}}{\left( \frac{k+1}{p}\right)! }
where
\tilde{\omega}(0,T)=\sum_{j=1}^{[p]} \left\|  \int dx^{\otimes j}-   \int  dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} .
We deduce
\| \Gamma_{s,t} \| \le \frac{2p}{\beta^2(1-2^{1-\theta})} \tilde{\omega}(0,T) C^{k+1} \frac{ \omega(s,t)^{(k+1)/p}}{\left( \frac{k+1}{p}\right)! }
with \theta= \frac{k+1}{p}. A correct choice of \beta finishes the induction argument \square

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