A Survey on exact analytical and numerical solutions of some S.D.E.s based on martingale approach and changing variable method

In this paper, we decide to represent analytical and numerical solutions for stochastic differentialequations, specially reputed and famous equations in pricing and investment ratemodels. By making martingale process from an arbitrary process in L2(R) space, we inferequations just with stochastic part (drift free). This method could be done by Ito productformula on initial process and an appropriate martingale process, then we compare simulatingmethod of arising this new equation with other simulating method like as E.M. andMilstein. Another suitable method is converting S.D.E.s to O.D.E.s whom we try to omitdiffusion part of stochastic equation. Afterwards, it could be solved by different numericalmethods like as Runge-kutta from fourth order. In this paper, we solve well known equationssuch as Gampertz diffusion and logistic diffusion by this method. Another powerful one ischange of variable method whom we could analysis and survey a well known group of stochasticequations like as special case of squared radial Langevin process, Cox-Ingersoll-Rossmodel and Ornstein-Uhlenbeck process. For numerical solution of these stochastic equations,we could apply wiener chaos expansion method whom we have described in other paper.

Keywords:

Numerical solution , Ito formula , Stochastic equations , Wiener chaos expansion