Uniqueness of a SDE with positivity constraint

We start by fixing some notation.

If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it should be clear from the context.

If $A=(a_{i,j})\in\mathcal M_{d,m}(\Bbb R)$, we define \begin{align*} &\|A\|_1^{(d,m)}:=\sup\{\|Av\|:v\in\Bbb S^{m-1}\}\\ %&\|A\|_2^{(d\times m)}:=\sqrt{\sum_{i,j}a_{i,j}^2}\;. \end{align*}

\begin{equation}\label{norms}%norms %c_{d,m}\|\cdot\|_1^{(d,m)}\le\|\cdot\|_2^{(d\times m)}\le C_{d,m}\|\cdot\|_1^{(d,m)}\;. \end{equation}

Take now $f:[0,L]\to\mathcal M_{d,m}(\Bbb R)$ and $h:[0,L]\to\Bbb R^N$; we define \begin{align*} \|f\|_{\infty,[0,L]}:=&\sup_{0\le t\le L}\|f(t)\|_1^{(d,m)}\;,\;\;\ \|f\|_{\lambda,[0,L]}:=\|f\|_{\infty,[0,L]}+\sup_{0\le s<t\le L}\frac{\|f(t)-f(s)\|_1^{(d,m)}}{|t-s|^{\lambda}}\\ \|h\|_{\infty,[0,L]}:=&\sup_{0\le t\le L}\|h(t)\|\;\;\;,\;\;\;\; \|h\|_{\lambda,[0,L]}:=\|h\|_{\infty,[0,L]}+\sup_{0\le s<t\le L}\frac{\|h(t)-h(s)\|}{|t-s|^{\lambda}} \end{align*} I know the document will be a little heavy with this notation, but it will be the key that will allow us to treat the problem when $(d>1)\vee (m>1)$.

We want to prove uniqueness for the solutions of the following $d$-dimensional stochastic differential equation \begin{equation}\label{sde}%sde x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\sigma(s,x_s)\,dW_s}_{=:z_t}+y_t \end{equation} where $y_t=\sup_{0\le s\le t}(z_s)^{-}$ is the regulator term which ensures that the positivity constraint is respected and the stochastic integral is a Young integral.

Now, $\xi_0\in\Bbb R_+^d:=\{(x_1,\dots,x_d)\in\Bbb R^d\;:\;x_i>0\;\;\forall i=1,\cdots,d\}$ is fixed, \begin{align*} &b:[0,L]\times\Bbb R^d\to\Bbb R^d\;\;\;\;,\\ &\sigma:[0,L]\times\Bbb R^d\to\mathcal M_{d,m}(\Bbb R) \end{align*} are measurable and bounded functions which satisfy \begin{equation}\label{hyp1}%hyp1 \|b(t,x)-b(t,y)\|\le K_0\|x-y\|\;\;\forall x,y\in\Bbb R^d,\;\forall t\in[0,L] \end{equation} \begin{equation}\label{hyp2}%hyp2 \|\sigma(t,x)-\sigma(t,y)\|_1^{(d,m)}\le K_0\|x-y\|\;\;\forall x,y\in\Bbb R^d,\;\forall t\in[0,L] \end{equation} \begin{equation}\label{hyp3}%hyp3 \|\sigma(t,x)-\sigma(s,x)\|_1^{(d,m)}\le K_0|t-s|^{\nu}\;\;\forall x\in\Bbb R^d,\;\forall s,t\in[0,L] \end{equation} where $\nu\in]\frac12,1]$ and $K_0>0$.

Hence, both $b$ and $\sigma$ are Lipschitz in space, moreover $\sigma$ is $\nu$-Holder continous in time.

Next we take a fractional Brownian motion $W$ of Hurst parameter $\gamma$ that is, $W\in\mathcal C^{\gamma}([0,L],\Bbb R^m)$ with $1/2<\gamma\le\nu$.

We will consider an arbitrary fixed trajectory of this FBM (i.e. $\omega$ is fixed and we mean $W=W(\omega)$).

Then it is proved that for every fixed $\lambda\in]\frac12,\gamma[$, the equation at the beginning admits a solution $x$ such that, a.s. $x(\omega)\in\mathcal C^{\lambda}([0,L],\Bbb R^d)$.

We summarize here the relations between the parameters considered above, in order to be clear: $$\frac12<\lambda<\gamma\le\nu\le1\;\;.$$

I would like to prove uniqueness for this SDE.

In the following we will consider $x^{(1)},x^{(2)}$ two $\lambda$-solutions of the SDE on $[0,L]$, writing $z^{(i)},y^{(i)}$ with the obvious meaning: $x^{(i)}=z^{(i)}+y^{(i)}\;\;i=1,2$.\ We will take $0\le T\le L$, which will be chosen conveniently later.

I used a classical technique: I used the $\sup$-norm, and I tried to show that $H_t:=\|x^{(1)}-x^{(2)}\|_{\infty,[0,t]}$ is zero for some $t>0$.

I skip all the passages, in order not to bother you, and I will go to the core of the problem: setting \begin{align*} A_t&:=at+bt^{\gamma}\\ B_t&:=ct^{\mu_{\beta}} \end{align*} where $\mu_{\beta}:=\gamma+\lambda(1-\beta)>1$, for some suitable $\beta\in[0,1[$, then I proved that $$H_t\le A_tH_t+B_tH_t^{\beta}$$ which is equivalent to $$H_t^{\beta}B_t\left(\frac{H_t^{1-\beta}}{\left(\frac{B_t}{1-A_t}\right)}-1\right)\le0$$ so the idea was to prove that there exist $T>0$ such that $$\frac{H_t^{1-\beta}}{\left(\frac{B_t}{1-A_t}\right)}-1>0\;\;\forall t\in[0,T].$$

Now, it's clear that $$\frac{B_t}{1-A_t}\sim_0t^{\mu_{\beta}}$$ thus I would like to study the behavior of $H$ near zero.

It is well known that our SDE admits solutions $x$, which are $\lambda$-Holder continuous; then the modulus of difference of such functions is again such; and taking the sup, the regularity stays the same or increases. But we must remember that we are studying the local behavior near $0$: how does $H$ behaves here? It's quite simple to show that $$\limsup_{t\to0^+}\frac{H_t}{t^{\lambda}}<+\infty$$ but this doesn't help me too much; it would be rather more useful to show that $$\liminf_{t\to0^+}\frac{H_t}{t^{\lambda}}>0\;\;\:$$ this last limit would allow me to conclude.

But maybe it is too much; and I conjectured that, using the fact that in RHS of the SDE a stochastic integral wrt a FBM appears, this would give a "constant irregularity" to my solution. I mean: $t\mapsto t^{\alpha}$ is $\alpha$-Holder on $[0,1]$, but the only bad point is $0$, otherwise it is always very regular, so paradoxically the possibility to have a smooth part, denies me the possibility to say anything about an arbitrary point, while, if we have -say- a FBM trajectory, we know that it is ALWAYS bad, in any point, but always with the same irregularity, and I thought that this maybe could help, since we are working locally!

Thus, hoping that the FBM driving the stochastic integral affects the solution with its irregularity (I don't have so much confidence with Young integral) I conjectured the following statement: $\forall \varepsilon>0\;\;\;\exists\delta>0$ such that $$t^{\lambda+\varepsilon}\le H_t\le t^{\lambda-\varepsilon}\;\;\;\forall t\in[0,\delta]$$ but I'm in trouble proving this last one. I feel it can be true, but I'm not able to prove it, for the moment.

However it would be interesting since it would allow me to conclude.