# Implicit Scheme for Cox-Ingersoll-Ross Model PDE

I am considering the PDE for the price of a bond $$V(r,t)$$ with maturity $$T$$ under the Cox-Ingersoll-Ross model,

$$V_t+\frac12\sigma^2rV_{rr}+\nu(\theta-r)V_r-rV=0\quad r>0, t\in(0,1)$$

with terminal condition $$V(r,T)=1$$ and $$\sigma$$, $$\nu$$ and $$\theta$$ are strictly positive constants. I have a few questions about the implicit scheme, which for brevity I will just express as $$V^{(n+1)}_j=a_jV^{(n)}_{j-1}+b_jV^{(n)}_j+c_jV^{(n)}_{j+1}$$ first.

1. I am trying to think of a suitable Dirichlet boundary condition for $$r\to\infty$$, or rather in this case $$r\to r_{\text{max}}$$. Intuitively, just as how $$V\to0$$ for $$S\to\infty$$ for the Black-Scholes equation, I am also thinking that $$V\to0$$ for $$r\to\infty$$ here. However, I am not sure how to justify it (the Black-Scholes has an analytic argument that I understand, but I have not found any literature about this CIR PDE).

2. Is it necessary for a right-sided first order finite difference at $$r=0$$ to replace the central difference? Intuition again tells me that this is a requirement for positivity of $$V$$ $$\forall$$ $$n$$, is this true? Are there further conditions should I be enforcing on $$\Delta t$$, $$\Delta r$$ and the constants?

3. I was told we can perform an upwind discretisation with right-sided (respectively left-sided) differences for $$r_n\leq\theta$$ (respectively $$r_n\geq\theta$$) to aid enforcement of positivity of $$V$$. I don't see that at all. What does the scheme look like and how does it ensure positivity?

(NB: I have shifted this question from MSE.)