A positivity preserving numerical method for stochastic R&D model

摘要

The stochastic research and development (R&D) model plays an important role in the growth rates of technological progress and capital accumulation. However, we can not obtain the explicit solution of stochastic R&D model. To explore a numerical approximate method preserving positivity for the R&D model with uncertainty from the population growth, we first construct an explicit Euler–Maruyama (EM) method and then verify it converges to the true solution with strong order 12(1−1p). But the EM method may lead to a negative approximation which is unrealistic. So, we establish the balanced explicit method (BEM) to overcome the defect of EM method and give sufficient conditions to preserve the positivity of BEM, then the convergence order of BEM is obtained. Finally, numerical simulations are carried out to verify our theoretical work.