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

With existing technology of Young differential equations for $x_{t}\in C^{\gamma}$, for the Picard iteration to go through one can prove existence/uniquence over $C^{\beta}$-solution space with $\gamma>\beta>1-\gamma$. For example, in "Stochastic differential equations driven by fractional Brownian motions" they show existence and uniqueness over the solution space $C^{\beta}$ with $\gamma>\beta>1-\gamma$. So one cannot use the existing Young technology for $\gamma<1/2$ since $\beta>1-\gamma>1/2>\gamma$.

But one can use RDEs see Proposition 6.12 "Rough Path Theory Lecture Notes Andrew L. Allan" where the author has to use RDEs in order to solve the equation $Y_t=\int Y_s dX_s$.

Having said that, here is an argument showing that only $Y\equiv 0$ can solve the above DE

$$Y_t=\int_{0}^{t} Y_s dX_s. $$

Suppose otherwise that there is some nonconstant solution $Y_{s}\in C^{\beta}$ and $x_{s}\in C^{\gamma}$ with $\beta+\gamma>1$ and $\gamma<1/2$. Using the lemma 2.3

we have

$$\frac{Y_t-Y_s}{(t-s)^{\beta}}=Y(s)\frac{X_t-X_s}{(t-s)^{\beta}}+\frac{\sum\sum...}{(t-s)^{\beta}}.$$

In the proof of prop 2.2, they show that the second term with the sums is bounded by $c (t-s)^{\gamma}(t-s)^{\beta}$ and so

$$\frac{Y_t-Y_s}{(t-s)^{\beta}}=Y(s)\frac{X_t-X_s}{(t-s)^{\beta}}+O((t-s)^{\gamma}).$$

However, since $\gamma<1/2$, that means $\beta>1/2>\gamma$ and so the first term on the RHS diverges whereas the LHS is finite, which is a contradiction. So we can only have zero solutions $Y\equiv 0$.