In the below attached text, I got the difference between both the probabilities and how they are affected by change in poisson parameter. But my qus. is in finite-time ruin prob. :- As Lembda increases, the prob. of finite-time ruin also increases. But, by increasing lembda, both the premium rate and the claim rate increases. Premium increases→Ruin prob. decreases Claim rate increases→Ruin prob. increases So, how do we know that the overall ruin prob. will increase? It means the rate of increasing premium < rate of increasing claim? If yes, can we prove it? Please anyone help. Thanks
Essentially what happens is that the whole process speeds up. So if we double lambda claims come in twice as fast and premiums come in twice as fast. So that means when the claim comes the premium is at the same point is was on the original version. So all that happens is that the process is faster - so ultimate ruin probability won't change as if you weren't ruined before you won't be now. However, since the process is sped up - then claims occur earlier - so if you were checking ruin until time 10, say, then before a simulation might have had ruin at time 16 but now has it at time 8 and so ruin in a particular time will increase. Above is an example of a simulation where I've doubled the lambda (red) so you can see that the ruin previously occurred about time 9 but now occurs at time 4.5 - hence the probability of ruin by time 5 has increased as at least one simulation more has ruin before 5 which didn't have it before.
