As part of a course in branching processes, the teacher gave the following definition of a branching process:
A stochastic process $X_t$ is called continuous-time branching process if there exists a sequence of positive functions $f_k(x,t)$, s.t.: $$ X_t = X_0 + \sum_{k=1}^{m} (k-1)Y_k\left[ \int_0^t f_k(X_s,s)ds \right]$$ For i.i.d. Poisson r.v. $\{Y_k[t]\}$.
This is the justification that I was given: We would like to define a branching process, in which every object has the same reaction rate. Let $\lambda_i(x,t)$ be the split rate to $i$ objects in the interval $[t,t+dt]$, and $p_i(n,x)$ be the probability for $i$ descendants. For each object, the chance for a splitting reaction into $i$ objects in the interval is $\lambda_i(x,t)dt + o(dt)$. If $X_t$ is the population size in time $t$, then for a given partition $\{t_i\}$ of $[0,t]$:
$$X_t - X_0 =\sum_{i=1}^{n} X_{t_i} - X_{t_{i-1}}$$
If we assume that $dt$ is small enough, s.t. $\lambda_i$ is constant, we get: $$X_t - X_0 =\sum_{i=1}^{n} \left( \sum_{k=1}^{m} (k-1)Y_k \left[ X_{t_i} \cdot \lambda_i(X_{t_i},t_i)dt\right] + o(dt) \right)$$ Where $Y_k[t]$ are independent Poisson random variables. Changing the summation order, we find: $$ X_t - X_0 =\sum_{k=1}^{m} (k-1) \left( \sum_{i=1}^{n} Y_k \left[ X_{t_i} \cdot \lambda_i(X_{t_i},t_i)dt \right] + o(dt) \right) \\ = \sum_{k=1}^{m} (k-1)Y_k\left[ \sum_{i=1}^{n} X_{t_i} \lambda_k(X_{t_i},t_i)dt + o(dt) \right] \\ \xrightarrow[dt\rightarrow 0]{} \sum_{k=1}^{m} (k-1) Y_k\left[\int_0^tX_s\lambda_k(X_s,s)ds\right] $$
I was trying to find this definition in other sources in the literature, but I couldn't find anything. Moreover, I feel like there are some missing gaps in some of the steps, such as the passage from the first equation to the second, how we managed to get the Poisson random variable outside the summation, and the integral over the branching process.
I would like to know if this definition for a continuous branching process is a standard one, and if so, where exactly in the literature is it found. I would also like to know if there's any reasoning behind this justification, or if the steps even make sense, because I don't really understand what's going on here.