Asymetric double exponential distribution $X \sim AL(p,\theta_1, \theta_2)$ has the following form of density function: $$ f(x) = p \theta_1 e^{−\theta_1x} \mathbb{1}_{x<0} + (1 − p) \theta_2 e^{\theta_2x} \mathbb{1}_{x\geq0} $$ I know the Inverse CDF method of simulation random variable, but in this case I think it is not possible, because CDF can not be presented in one simple form, so my question is:
How can I simulate this distribution?
Well, it should be fairly obvious, since $$\Pr[X \ge 0] = \int_{x=0}^\infty f_X(x) \, dx = 1-p,$$ and $$\Pr[X < 0] = p.$$ Thus to simulate realizations of $X$, all you need to do is simulate a $\operatorname{Bernoulli}(p)$ variable, and based on the value of this variable, simulate either an $-\operatorname{Exponential}(\theta_1)$ or $\operatorname{Exponential}(\theta_2)$ variable.
Incidentally, you have the signs reversed if $\theta_1, \theta_2 > 0$. Your present parametrization is valid only if $\theta_1, \theta_2 < 0$.