Consider a sequence of parametrized SDEs :
$$X^{a}_t = z + \int_0^t b(a,s,X^a_s)ds+\int_0^t\frac{\sigma(a,s,X^a_s)}{1+{\bf 1}_{\{b(a,s,X^a_s)>0\}}}dW_s,\quad \forall t\ge 0,~~~~~~~~~~~~~~~~~~~~~(\ast)$$
where $z\in\mathbb R$ and $b,\sigma: \mathbb R\times \mathbb R_+\times \mathbb R\to\mathbb R$ are Lipschitz and bounded. Assume further $\inf_{(a,t,x)}\sigma(a,t,x)>0$. My questions are as follows :
Under which conditions $(\ast)$ admits a unique (weak) solution?
Is there some subset $A\subset\mathbb R$ s.t. $A\ni a\mapsto Law(X^{a})$ is continuous with respect to the weak convergence?
