# Existence/uniqueness of the solution to a two-dimensional SDE

Let $$D$$ be the space of functions on $$\mathbb R_+$$ that are right-continuous with left limits. Consider two measurable functions $$a,b: \mathbb R_+\times D\to [0,1]$$ that are non-anticipative, i.e.

$$a(t,f)=a(t,f^t) \quad\mbox{and}\quad b(t,f)=b(t,f^t),\quad \forall (t,f)\in \mathbb R_+\times D,$$

where $$f^t\in D$$ is defined as $$f^t(s):=f\big(\min(t,s)\big)$$. Given two independent Brownian motions $$B,W$$ starting at zero, consider the following two-dimensional SDE:

$$\begin{eqnarray} X_t &=& x + B_t + \int_0^ta(u,X)du - \alpha X_{\sigma} {\bf 1}_{\{\sigma\le t\}},\quad \forall 0\le t\le \tau \\ Y_t &=& y + W_t + \int_0^tb(u,Y)du - \beta Y_{\tau} {\bf 1}_{\{\tau\le t\}},\quad \forall 0\le t\le \sigma \\ \end{eqnarray}$$

where $$\tau:=\inf\{t\ge 0: X_t \le 0\}$$, $$\sigma:=\inf\{t\ge 0: Y_t \le 0\}$$, and $$\alpha, \beta \in (0,1)$$, $$x,y>0$$ are constants. Does the above SDE admit a unique solution $$(X_t,Y_t)_{0\le t\le \max(\tau,\sigma)}$$?

PS : The existence is straightforward. Construct two independent processes $$\bar X, \bar Y$$ as the solution of

$$\begin{eqnarray} \bar X_t &=& x + B_t + \int_0^ta(u,\bar X)du,\quad \forall t\ge 0 \\ \bar Y_t &=& y + W_t + \int_0^tb(u,\bar Y)du,\quad \forall t\ge 0. \\ \end{eqnarray}$$

Define by $$\bar \tau, \bar\sigma$$ the first hitting time at zero of $$\bar X, \bar Y$$. Then we define $$X,Y$$ by distinguishing the following two cases

1. If $$\bar \tau\le \bar\sigma$$, then $$X_t:=\bar X_t$$, $$Y_t:=\bar Y_t$$ for $$t<\bar \tau$$ and for $$t\ge \bar\tau$$, $$Y_t$$ solves

$$Y_t=(1-\beta)Y_{\bar\tau-}+ (W_t-W_{\bar\tau})+\int_{\bar\tau}^tb(u, Y)du$$

1. If $$\bar \sigma\le \bar\tau$$, then $$Y_t:=\bar Y_t$$, $$X_t:=\bar X_t$$ for $$t<\bar \sigma$$ and for $$t\ge \bar\sigma$$, $$X_t$$ solves

$$X_t=(1-\alpha)X_{\bar\sigma-}+ (B_t-B_{\bar\sigma})+\int_{\bar\sigma}^ta(u, X)du.$$

Is there any mistake in my construction? If not, is this the unique solution?