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Implicit Taylor methods for stiff stochastic differential equations. (English) Zbl 0983.65007

Three implicit numerical methods are presented for approximating the solution of a stiff Itô stochastic differential equation of the form \[ dy(t)= f(y(t)) dt+ \sum^d_{j=1} g_i(y(t)) dW_j(t),\quad y(t_0)= y_0, \] where \(W_j(t)\), \(j= 1,\dots, d\), are independent Wiener processes.
The first, called the implicit Euler-Taylor method (IET), is shown to have strong convergence of order .5 and a superior region of mean square stability when compared to its corresponding explicit and semi-explicit methods.
The second called the implicit Milstein-Taylor method (IMT), is shown to have strong convergence of order 1.0 and a superior region of mean square stability when compared to its corresponding explicit and semi-explicit method, but a slightly inferior region as a compared to the IET method.
The third, called the implicit 1.5 Taylor, is shown to have strong convergence of order 1.5, but an inferior region of mean square stability when compared to its corresponding semi-explicit method and a very inferior region when compared to the IET and IMT methods.
The paper concludes by giving numerical results for two test equations.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
34F05 Ordinary differential equations and systems with randomness
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)

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