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Stable strong order 1.0 schemes for solving stochastic ordinary differential equations. (English) Zbl 1259.65002

A class of stable numerical methods (SSO1), an extension of the balanced method and the Milstein method, for solving systems of stiff Itô stochastic differential equations of the form \[ dX(t)= f(X(t))\,dt+ \sum^d_{i=1} g_i(X(t))\,dW^i(t),\quad X(t_0)= X_0, \] is presented. A theorem is proved that gives conditions under which strong convergence of order 1 to the exact solution is achieved. The theorem is then extended to establish conditions under which a generalized balanced scheme has strong order of convergence \(p\). A procedure for choosing the parameters in the SSO1 method to minimize global error is given. An examination of asymptotic stability is performed that shows that the SSO1 method surpasses the Milstein method, semi-implicit Milstein methods, and the balanced method.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
65L04 Numerical methods for stiff equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations

Software:

S-ROCK; RODAS
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Full Text: DOI Link

References:

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