Adaptive Analysis Of Solutions For Two Kinds Of Stochastic Systems

Stochastic effects are ubiquitous in real life and it is crucial to choose appropriate equations to describe such effects.Stochastic differential equations are often used as practical models to study such problems.Stochastic differential equations are differential equations with random perturbation parameters,and such models have been widely used in various fields such as economics,biology,and physics.In this paper,stochastic capital model and stochastic biological population model are studied in a systematic way respectively.It mainly includes:The convergence study of the numerical solution of the stochastic scale structure system with resource term:As one of the important components of biological research,population studies play a crucial role in the process of biological development.Most populations are affected by stochastic factors in the process of growth and reproduction,so considering the effects of stochastic factors in the study of population problems will make the study more relevant to reality.A category of stochastic scale systems with special resource terms is considered,and the main purpose is to analyze the convergence of the numerical solution of the system.First,a semi-implicit Euler numerical method is used to construct the numerical solution of the discrete model.Then,the mean square convergence of the numerical solution of the system is discussed using It?’s Lemma.Finally,the stochastic population model with scale structure is simulated numerically according to the characteristics of the discrete system,and the reliability of the numerical method is verified.Numerical analysis of stochastic capital systems with time lag and diffusion terms:Capital systems are widely used in the economic field.The article constructs a more complete capital system with stochastic factors by considering the effects of time lags and diffusion on the system based on the study of stochastic capital systems with Poisson jumps.A class of stochastic age capital systems with time lag and diffusion terms is considered,and the primary objective is to ensure the positivity and mean square dispersion of the capital system solutions using suitable numerical methods.First,we need to guarantee the positivity of the real solution due to the practical significance requirements of the capital,but there exists no means to guarantee the positivity of the solution with the numerical solution constructed using the semi-implicit Euler method.In order to solve this problem,the balanced implicit idea is introduced and it is proved that the balanced implicit method ensures the positivity of the solution under certain assumptions.Secondly,under certain assumptions,it is proved that the numerical solution obtained using the balanced implicit method converges to the real solution of the system.Finally,sufficient conditions for the mean square scattering of the capital system and the balanced implicit method are given and the mean square scattering of the system is verified using numerical simulations.