Proving Existence and Uniqueness for Cox-Ingersoll-Ross SDE.

Question

The Cox-Ingersoll Ross SDE is:

$dr_t=a(b-r_t)dt+\sigma\cdot \sqrt{r_t}dB_t$. I would like to know how to prove existence and uniqueness and that the solution is positive.

Øksendal has this result(I simplify it to one dimension):

Let $T>0$ and $b(\cdot,\cdot): [0,T]\times\mathbb{R}\rightarrow R, \sigma(\cdot,\cdot): [0,T]\times\mathbb{R}\rightarrow \mathbb{R}$ be measurable functions satisfying:

$$|b(t,x)|+|\sigma(t,x)|\le C(1+|x|),$$

for some $C$. And also

$$|b(t,x)-b(t,y)|+|\sigma(t,x)-\sigma(t,y)|\le D|x-y|,$$

for some $D$. Let $Z$ ve a random variable which is independent of the sigma-algebra $F_\infty$ generated by $B_s$ and such that

$$E[Z^2]<\infty.$$

Then the stochastic differential equation

$$dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t, X_0=Z,$$

has a unique t-continuous solution $X_t(\omega)$ with the property that $X_t(\omega)$ is adapted to the filtration $\mathcal{F}_t^Z$ generated by $Z$ and $B_s$, $s\le t$ and

$$E\left[\int_0^T |X_t|^2 dt\right]<\infty.$$

Can we use this result to show that the CIR SDE has an unique positive result? The problem is that the growth conditions are not satisfied near zero. An idea is to look at the SDE:

$dr_{t,\epsilon}=a(b-r_{t,\epsilon})dt+\sigma\cdot \sqrt{\max(r_{t,\epsilon},\epsilon)}dB_t,$

from what I see this function satisfies the growth and Lipschitz continuity condition for every $\epsilon$ bigger than zero. If we let $\epsilon_n$ be a sequence of positive real numbers converging to zero we get a sequence of processes $r_{t,\epsilon_n}$. But do we know if these processes converges in some way to the process we want?, and if they converge in some way to a process, is the process an Itö-process that satisfies the CIR SDE?

Answer 1

I'm not sure if your suggested approach works, but let me propose an alternative method of showing existence and pathwise uniqueness of solutions to the CIR SDE:

Theorem (Yamada and Watanabe) Let $\rho $ be a strictly increasing continuous function with $\rho (0)=0$ and $\int_0^r \rho (t)^{-2} dt =\infty$ for all $r > 0$; further let $\kappa$ be strictly increasing and concave with $\kappa (0) =0$ and $\int_0^r \kappa (t)^{-1}dr = \infty$ for all $r > 0$. Then if $|\sigma (x) - \sigma (y)| \leq \rho (|x-y|)$ and $|b(x) - b(y)| \leq \kappa (|x-y|)$ hold for all $x,y$ then existence and pathwise uniqueness holds for $$dX_t = b(X_t)dt + \sigma (X_t) dB_t$$

Notice that these conditions are satisfied for the CIR process, giving you the result you need.