Near-optimality Of Stochastic Population Systems

As an application of the stochastic differential equations in Biology, stochastic population system has received a great deal of attentions. There are many random factors in the population living environment, and they are likely to change the population’s amount. For solving practical problems effectively, control-ling the development of populations reasonably, it is necessary to select the appropriate control variable and establish performance indicators to study the optimal control of the stochastic population systems. However, for the optimal control it needs strict conditions relatively, and also not any one has its own optimal control. So, in this paper, we study the near-optimality of stochastic population systems. Main contents are as follows:1. The theorem, definition, lemma, inequalities that will be used in the paper are given, especially relevant to Brown motion, Poisson jumps, Markov switching.2. With the adjoint equations and Hamiltonian functions, using Itd’s formula%Barkholder-Davis-Gundy’s inequality and Ekeland’s variational principle and so on, we study the near-optimal control prob-lem of stochastic population systems with Poisson jumps, and then give the necessary and sufficient conditions.3. For better description of the population systems, we introduce the Fractional Brownian motion and Markov switching into the stochastic population systems, and discuss the near-optimality of the stochastic population systems. At last, using the maximum principle on the Hamiltonian function we give the the necessary and sufficient conditions for near-optimality control problem of the systems.