Correlated Poisson Distribution $X_1$ and $X_2$ are discrete stochastic variables. They can both be modeled by a Poisson process with arrival rates $\lambda_1$ and $\lambda_2$ respectively.
$X_1$ and $X_2$ have a constant correlation $\rho$.
Is there an analytic equation that describes the probability density function: 
$P(X_1= i,X_2= k)$ 

You haven't given enough information to say what their joint distribution is.  But maybe the simplest possibility is this:
\begin{align}
X_1 & = Y_1 + Y_2 \\
X_2 & = Y_2 + Y_3
\end{align}
where $Y_1,Y_2,Y_3$ are independent Poisson-distributed random variables with expectations $\mu_1,\mu_2,\mu_3$, and of course $\mu_1+\mu_2=\lambda_1$ and $\mu_2+\mu_3=\lambda_2$.  Recall that the variance of a Poisson-distributed random variable is the same as its expected value.  Then the correlation is $\rho$ if
$$
\operatorname{cov}(X_1,X_1) = \rho\sqrt{\operatorname{var}(X_1)\operatorname{var}(X_2)}=\rho\sqrt{\lambda_1\lambda_2}.
$$
The covariance is
$$
\operatorname{cov}(X_1,X_1) = \operatorname{cov}(Y_1+Y_2,Y_2+Y_3) = \operatorname{var}(Y_2)=\mu_2. 
$$
From there you can think about possible values of $\mu_1,\mu_2,\mu_3$ as functions of $\lambda_1,\lambda_2$.
