# Uniqueness of solutions of Young differential equations

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 $$\beta<1/2$$?

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?

• "Maybe I am missing something here, but the proof definitely does not follow immediately from your comment." It does. Let $t>0$. Choose small $\varepsilon>0$ and consider the last moment $t_1\in[0,t]$ when $|Y_t|\le\varepsilon$. Then $|Y_s|\ge\varepsilon$ on $[t_1,t]$, so $X\in Hol_\alpha([t_1,t])$ and then $Y_s=Y_{t_1}\exp(X_s-X_{t_1})$ for $s\in[t_1,t]$. In particular $|Y_t|\le\varepsilon\exp(2\|X\|_{C([0,t])})$. But $\varepsilon>0$ is arbitrarily small! Jul 27, 2020 at 12:50
• @ThomasKojar What do you mean I cannot impose a different regularity? Of course I can, this is the whole point of the question. If we know in advance that $Y$ is very regular, show that $Y$ is identically zero (or provide a counter-example).
– Oleg
Jan 30 at 21:48
• @ThomasKojar Let me rephrase my question. We are given two processes. $Y$ which is very regular and $X$ which is less regular. It is known that $dY_t=Y_td X_t$ (in the usual ODE, not RDE sense, all the integrals are well-defined Young integrals). Question: is it true that $Y\equiv0$?
– Oleg
Jan 30 at 21:51
• In other words: I am asking for uniqueness only among $C^\alpha$ solutions (not among all solutions) for which we can always define this as a Young integral. Is it more clear now?
– Oleg
Jan 30 at 23:34
• @ThomasKojar I really do not understand what you do not understand in my question. We are given two processes $Y$ and $X$, $Y$ is regular $X$ is not. They solve a certain equation. Question: does this mean that $Y=0$ or there is another regular process $\widetilde Y$ such that $d\widetilde Y=\widetilde Y dX$. You are asking a different question, but which part in my question you do not understand? I am talking here about standard ODEs not RDEs. You are trying to answer a different question.
– Oleg
Jan 30 at 23:39

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 $$Y\equiv 0$$.

• The answer is totally unrelated to the question. Jan 30 at 12:51
• @user479223 ok I updated the answer. Thank you for your help. Jan 31 at 1:41