Master equation for reproduction and mutual annihilation process

I was solving some exercises regarding the Master Equation and couldn’t solve the following problem.

Consider a population with individuals $A$. This population can suffer the following processes:

i) Reproduction: $A xrightarrow{text{$sigma$}} A + A$

ii) Mutual annihilation: $A + A xrightarrow{text{$lambda$}} 0 $

where $sigma$ and $lambda$ are rates of evolution.

My attempt

I started by writing the transition rates $omega_{n,n’}$ such as described in Reichl – A modern course in statistical physics, page 260. Hence, for process i) I got

$$omega_{n,n+1} Delta t = sigma n Delta t = sigma_n Delta t$$

while for the second

$$omega_{n,n-2} Delta t = frac{1}{2} n(n-1) lambda Delta t = lambda_n Delta t $$

which yielded the master equation

$$partial_{t}P(n,t) = -(lambda_n+sigma_n)P(n,t) + sigma_{n-1}P(n-1,t) + lambda_{n+2}P(n+2,t) $$

I was wondering: is this correct? I am unable to check this result.