In page 1349 or Section 2.1 of "Duffie, D., Pan, J., & Singleton, K. (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica, 68(6), 1343-1376" an affine jump-diffusion (AJD) is defined as a Markov process such that: $dX_t=\mu(X_t)dt+\sigma(X_t)dW_t+dZ_t$, where $W$ is a Brownian motion, for the purpose of this question we can consider $\mu$ and $\sigma$ as constant, but $Z$ is defined as a compound Poisson process whose jumps are iid with distribution $\nu$ and jump times conditional on the path of $X$ "follow a Poisson process with time-varying intensity $\{\lambda(X_s):0\le{s}\le{t}\}$".
If I've understood correctly the statement in quotation marks means that $Z_t=\sum_{i=1}^{N_t}{Y_i}$ where $Y_i$ (jumps) are iid with distribution $\nu$ and $\{N_t\}_{t\le{0}}$ is a nonhomogeneous Poisson process such that:$$N_t|(X_s)_{0\le{s}<t}\sim{Pois}(\int_{0}^t{\lambda(X_u)du}).$$
Now, how can I know if such a differential equation has a solution (analitical or numerical)?
Then, if I would like to simulate such a process, where should I begin from? Calling $N_t$ the jump times at time $t$, I can't just simulate $W_t$ and $N_t$ alone since $N_t$'s distribution depends on $X$; but for the same reason I can't simulate first $X$ and then $N_t$ since $X$'s equation depends on $N_t$.
Edit: In Section 1.3 of "Øksendal, B. K.; Sulem Agnès. Applied Stochastic Control of Jump Diffusions, Third.; Universitext; Springer: Cham, Switzerland, 2019" I can find a Theorem about "Existence and Uniqueness of Solutions of Lévy SDEs" which is a general case of an affine jump diffusion I defined before. The problem of simulation is not addressed in this book and it is still open.
To simulate the solution, you can use time-discretization scheme, for example, explicit Euler scheme. For $N\ge 1$, set $t_n^N = n/N$, $n\ge 1$, start with $X^N(0) = 0$ and define recursively $$ X^{N}(t_{n+1}^N) = X^{N}(t_n^N) + \mu\big(X^{N}(t_n^N)\big)\cdot \frac1N + \sigma \big(X^{N}(t_n^N)\big)\cdot \big(W(t_{n+1}^N) - W(t_n^N)\big) + \xi^N_n, $$ where $\xi_n^N$ is a random variable which has distribution $\nu$ (and is independent of everything else) with probability $\frac{\lambda\left(X(t_n^N)\right)}N\wedge 1$ and equals $0$ with complementary probability.