Expected value and variance for a biased 1d random walk with a boundary condition

Question

I’m working on a population genetics problem. Let’s say that N independent elements start with a value of 1 copy. At each generation they can either:(i) lose a copy (-1) with a probability nu;(ii) suffer a rearrangement with a probability theta that can lead to lose a copy (-1) or gain a copy (+1) with equal chance of either outcome; or (iii) stay the same with probability 1-nu-theta. If an element loses all copies (reaches value 0) then it stays like that for good. Other than that, the outcome of each generation does not affect probabilities down the road. I see this as a biased 1d random walk with an absorbing boundary condition at 0. I need expressions for the expected value and the variance of the sum of all N elements at generation g, as a funtion of N, g, theta and nu. I can work out the model without the boundary condition. From simulations, this seems good enough for small values of nu and theta compared to 1/Ng, but it breaks down for larger values. I guess that this is because as more elements reach 0 the stop contributing to the mean and the variance. I’ve been reading everything I can about this, starting with Feldman, but i don’t know how to use the “gambler’s ruin” concepts to refine the expected value and variance. Can anyone give some guidance?