Research On The Numerical Solutions And Its Applications For Stochastic Systems With Time Delays Under Poisson White Noise Excitations

Stochastic systems with time delays are mainly described by the sotchasic delay differential equations (SDDEs). Stochastic systems with delays are becoming increasingly used nowdays in dfferent fields, such as, economics, physics, medicine, ecology and so on. However few analytical solutions can be obtained for SDDEs, thus, developing appropriate numerical methods for SDDEs is interesting topic both in theory and in applications.In recent years, many scholars studied the approximate schemes for SDDEs and have achieved fruitful results. However, the study on the numerical schemes for stochastic delay differential equations with jumps (SDDEwJs), stochastic delay integro-differential equations (SDIDEs) and neutral stochastic delay differential equations with jumps (NSDDEwJs) has just begun. In this paper, we investigate the convergence and mean square stability of the numerical methods for these three types stochastic delay differential equations,at the same time we apply these stochachtic systems to biology and obtained the following results:1. For the SDDEs, Poisson white noise is added to this class equation to obtain a class of SDDEwJs. First, the definition of mean-square stability of numerical methods for SDDEwJs is established, and then the sufficient condition of mean square stability of the Euler-Maruyama method for SDDEwJs is derived, finally a class scalar test equation is simulated, the numerical experiments verify the results obtained from theory.2. The convergence and mean square stability of the numerical approximation of solutions for linear stochastic delay integro-differential equations (SDIDEs) was studied. Split-step backward Euler (SSBE) method for solving linear stochastic delay integro-differential equations is derived. It is proved that the SSBE method is convergent with strong order 12 in the mean-square sense. The condition under which the SSBE method is mean-square stable (MS-stable) is obtained. At last, the numerical experiments illustrate the validity and correctness the method.3. For the neutral stochastic functional differential equations (NSFDEs), Poisson white noise is added to this class equation to obtain a class of neutral stochastic functional differential equations with Poisson jumps (NSFDEwJs). First, the existence and uniqueness of solutions for NSFDEwJs is studied, then the Euler-Maruyama method for neutral stochastic delay differential equations with Poisson jumps (NSDDEwJs) is developed, and last we prove that the numerical solutions will converge to the true solutions for NSDDEwJs with strong order 1/2 under the local Lipschitz condition.4. The stochastic delay differential equations with jumps and stochastic delay integro-differential equations are applied to biological population dynamics. Using the numerical methods in this paper, we simulate the response of the systems and found that the stability and Hopf bifurcation point of the biological populations.