Let $s\ne t$ , How can I find $Cov(N_t,N_s)$?

Question

Let $N_t$ be a poisson proces with intensity $\lambda>0$ , I want to show $$\rho=\sqrt{\frac{\min\{s,t\}}{\max\{s,t\}}}$$ How Can I do it. Thanks

Answer 1

• If $s<t$

$$\begin{align} & \mathbb{E}\left[ {{N}_{t}}{{N}_{s}} \right]=\mathbb{E}\left[ {{N}_{t}}({{N}_{t}}-{{N}_{s}})+N_{s}^{2} \right] \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\mathbb{E}\left[ {{N}_{t}}({{N}_{t}}-{{N}_{s}}) \right]+\mathbb{E}\left[ {{N}_{t}}^{2} \right] \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\mathbb{E}\left[ {{N}_{t}} \right]\mathbb{E}\left[ {{N}_{t}}-{{N}_{s}} \right]+\mathbb{E}\left[ {{N}_{t}}^{2} \right] \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\lambda }^{2}}s(t-s)+\lambda s+{{\lambda }^{2}}{{s}^{2}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\lambda }^{2}}st+\lambda s \\ \end{align}$$ therefor $$\operatorname{Cov}({{N}_{t}},{{N}_{s}})=\mathbb{E}\left[ {{N}_{t}}{{N}_{s}} \right]-\mathbb{E}\left[ {{N}_{t}} \right]\mathbb{E}\left[ {{N}_{s}} \right]={{\lambda }^{2}}st+\lambda t-{{\lambda }^{2}}st=\lambda t$$

• Similarly, if $t<s$ $$\operatorname{\operatorname{Cov}}({{N}_{t}},{{N}_{s}})=\lambda s$$ We have $$\rho =\frac{\operatorname{Cov}({{N}_{t}},{{N}_{s}})}{\sqrt{\operatorname{Var}({{N}_{t}})\operatorname{Var}({{N}_{s}})}}=\frac{\lambda \min \{s,t\}}{\lambda \sqrt{st}}=\frac{\min \{s,t\}}{\sqrt{st}}$$ in other words $$\rho =\left\{ \begin{align} & \frac{s}{\sqrt{st}}=\sqrt{\frac{s}{t}}\,\,\,\,,\,\,\,\,\,s<t \\ & \frac{t}{\sqrt{st}}\,=\sqrt{\frac{t}{s}}\,\,\,\,\,,\,\,\,\,\,t<s \\ \end{align} \right.$$ finally $$\rho =\sqrt{\frac{\min \{s,t\}}{\max \{s,t\}}}$$