The Truncated Euler-maruyama Scheme With Positive Preserving Properties

The numerical solution of stochastic differential equations has been studied by many scholars.In recent years,there is a new numerical method truncated Euler Maruyama(EM)method.The truncated EM method can guarantee convergence for highly nonlinear equations.There are many models in finance,biology and other fields.Only when the solution of stochastic differential equation is positive,can it be of practical significance.However,the truncated EM method can not guarantee that the numerical solution of the equation is positive for these models.In this paper,an explicit positive preserving numerical method is adopted.To be more exact,it is an explicit logarithmic truncated EM method.In order to study the positivity of numerical solutions of more general stochastic differential equations,this paper starts with the partially truncated EM method.The main objective in the study is how to construct the numerical approximation of stochastic differential equation(SDE)when it has a positive solution,so that the numerical solution remains positive.In this paper,one-dimensional stochastic differential equations are considered.In order to obtain the positive result,firstly,a new stochastic differential equation is obtained by logarithmic transformation of the coefficients of the stochastic differential equation.Combined with Feller condition,the existence and uniqueness of the solution of the new stochastic differential equation is obtained.Secondly,the coefficient of the new stochastic differential equation is divided into the part that satisfies the linear growth condition and the part that does not satisfy the linear growth condition In order to ensure the convergence of the numerical solution,the exponential integrability of the numerical solution and the analytical solution of the new stochastic differential equation are further proved under the condition that the numerical solution of the new stochastic differential equation converges to the real solution.Finally,combined with the related properties of logarithmic transformation,the result that the numerical solution of the original stochastic differential equation keeps the positive analytical solution is obtained.