Abstract
In this paper we are concerned with numerical methods to solve stochastic differential equations (SDEs), namely the Euler-Maruyama (EM) and Milstein methods. These methods are based on the truncated Ito-Taylor expansion. In our study we deal with a nonlinear SDE. We approximate to numerical solution using Monte Carlo simulation for each method. Also exact solution is obtained from Ito’s formula. To show the effectiveness of the numerical methods, approximation solutions are compared with exact solution for different sample paths. And finally the results of numerical experiments are supported with graphs and error tables.
Highlights
Until recently, many of the models ignored stochastic effects because of difficulty in solution
Note that g (X(ti)) is differentiation of g(X(ti)), and if the type of stochastic differential equations (SDEs) is an additive noise SDE, the Milstein method leads to the Euler-Maruyama method
We say from our graphs that, if we minimize the stepsize dt because of dt = 1/N, we obtain more closed approximation to exact solution with the Milstein method compared with the Euler-Maruyama method
Summary
Many of the models ignored stochastic effects because of difficulty in solution. Enlarging the increments of smooth functions of Ito processes, it is beneficial to have a stochastic expansion formula with correspondent specialities to the deterministic Taylor formula. We can produce the numerical integration scheme for the SDE from Ito-Taylor expansion (19) with a time discretization 0 = t0 < t1 < · · · < tn < · · · < tN = T of a time interval [0,T] as follows: X(ti+1) = X(ti) + f X(ti) t + g X(ti) Wi g 2. If we truncate Ito’s formula of the stochastic Taylor series after the first order terms, we obtain the Euler method or Euler-Maruyama method as follows:. If we truncate the stochastic Taylor series after second order terms, we obtain the Milstein method as follows:. Note that g (X(ti)) is differentiation of g(X(ti)), and if the type of SDE is an additive noise SDE, the Milstein method leads to the Euler-Maruyama method
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