I am trying to solve the following one-dimensional PDE $$\partial_tv(t,x)+b(x)\partial_xv(t,x)+\frac{1}{2}\sigma^2(x)\partial_x^2(t,x)+(v-2v^{-1})(t,x)=0 \quad \text{ for } t \in (0,T)$$ with the boundary condition $$v(T,\cdot)=c,$$ where $c\in \mathbb{R}$.
The problem is that the function $x \mapsto x-2x^{-1}$ is only locally Lipschitz continuous. Is it still possible to solve the PDE numerically by simulating the corresponding backward SDE?
Thanks in advance!