The Research On Theory And The Application Of Branching Processes

Branching processes is one of extremely important stochastic process ,it is very widely used . There are many monographs overseas, but the domestic elaboration about branching processes is seen little of. So it is very important to put forward the systematic discussion and in-depth research.In this dissertation, We studied the definition,probability generating function,mean,variance,extinction probability and instability about classical branching processes. Then we introduced the linear controlled branching processes and some related problems in detail. we studied instability of the linear controlled branching processes, the concrete conclusions are:1 for the linear controlled branching processes {Z_n, n∈N₀}:is proved.2 for the linear controlled branching processes stopped at zero {W_n, n∈N₀}:is proved.we studied k times controlled branching processes, put forward probability generating function of the total number of all members up to n th generation.Finally, we studied application of branching processes in population genetics, its model is established in this paper, we get the extinction probability of mutant gene.