Uniqueness of solutions of Young differential equations

Question

Consider the following one dimensional Young differential equation: \begin{align*} &Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1];\\ &Y_0=0. \end{align*}

Here the driving process $X$ is a bounded functions $[0,1]\to\mathbb{R}$, which is $\beta$-Holder with $\beta<1/2$.

If $Y$ is an $\alpha$-Holder function, $\alpha>1-\beta$, then this equation is well defined (because the integral then becomes just the Young integral).

Question: how to prove that the only solution to this equation in the class of $\alpha$-Holder functions is $Y\equiv0$?

Warning: note that $\beta<1/2$! If $\beta>1/2$, then this result is standard, but what to do if $\beta<1/2$?

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Failed solution attempt

Denote by $[X]_{\beta,[0,T]}$, $[Y]_{\alpha,[0,T]}$ the corresponding Holder norms of $X$ and $Y$ on the interval $[0,T]$, respectively. Then the standard inequality for the Young integral implies $$ |Y_t-Y_s-Y_s(X_t-X_s)|\le C[X]_{\beta,[0,T]}[Y]_{\alpha,[0,T]} (t-s)^{\alpha+\beta},\quad s,t\in[0,T]. $$ This in turn leads $$ |Y_t-Y_s|\le C[X]_{\beta,[0,T]}[Y]_{\alpha,[0,T]} (t-s)^{\alpha+\beta}+|Y_s|\,|X_t-X_s|, $$ and thus $$ [Y]_{\beta,[0,T]}\le C[X]_{\beta,[0,T]}[Y]_{\alpha,[0,T]} T^{\alpha}+[X]_{\beta,[0,T]}\sup_{r\in[0,T]}|Y_r|. $$

However, because $\beta<\alpha$, the last inequality gives us nothing (we are estimating a smaller norm by a larger norm). The iteration over $T$ also seems hopeless. So what to do?

Answer 1

We will follow the Lemma 8.10. (Rough Gronwall) and Proposition 8.12 from "a course in rough paths" but modify them for this particular setting of studying

$$Y_{t}=Y_{0}+\int^{t}_{0} Y dX,$$

for $Y\in C^{\alpha},X\in C^{\beta}$ where $\alpha+\beta>1$ but $\beta<\frac{1}{2}$ and $\alpha>\frac{1}{2}$.

The Lemma 8.10. (Rough Gronwall) comes verbatim with no changes

Assume $Y\in C([0,1])$, $r<1$ and $$|Y_{s,t}|\leq M ||Y||_{\infty;[s,t]}|t-s|^{r},(*)$$ for $0\leq s< t\leq 1$. Then there exists $c=c_{r}<\infty$ such that $$||Y||_{\infty;[0,1]}\leq cexp(cM^{1/r})|Y_{0}|.$$

So it suffices to prove (*). From a slight variation of the proof of Proposition 8.12 we have the bound

$$|Y_{s,t}|=|\int_{s}^{t}Y_{r}dX_{r}|\leq c||Y||_{\infty;[s,t]}||X||_{\beta;[s,t]}|t-s|^{\beta}\leq c||Y||_{\infty;[s,t]}|t-s|^{\beta},$$ where we used that $\beta<\frac{1}{2}\leq \alpha$ and so $||Y||_{\alpha;[s,t]}\leq ||Y||_{\beta;[s,t]}$. So the result follows from here.