The composite Euler method for stiff stochastic differential equations (Q5939881)

A composite Euler method is presented for solving stiff stochastic differential equations of the form NEWLINE\[NEWLINEdy(t)= f(t, y(t)) dt+ \sum^l_{j=1} g_j(t, y(t)) dW_j(t),NEWLINE\]NEWLINE where \(W_j(t)\) is a Wiener process.NEWLINENEWLINENEWLINEConsidering first the case where \(l=1\), the semi-implicit Euler method and the implicit Euler method are combined to arrive at the composite Euler method NEWLINE\[NEWLINEy_{n+1}= y_n+ f(t_{n+1}, y_{n+1}) h+ [\lambda_n g(t_n, y_n)+ (1- \lambda_n) g(t_{n+1}, y_{n+1})] \Delta W_n,NEWLINE\]NEWLINE where \(\lambda_n\in [0,1]\) is selected a new at each step. Criteria for selecting \(\lambda_n\) are delineated which lead to two versions of the composite Euler method. \(MS\)-stability, \(T\)-stability, and convergence are studied for these versions. Numerical results for three test equations are presented which attest to the sucess of this approach. The paper concludes by generalizing the method to arbitrary \(l\), analyzing its stability properties, and providing numerical results for an equation where \(l=2\).