Convergence Results for SDEs by Bseries

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Convergence Results for SDEs by B-series

• Kevin Burrage
• Pamela Burrage
• Mathematics Department
• University of Queensland
• kb_at_maths.uq.edu.au
• pmb_at_maths.uq.edu.au

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• Deterministic case
• Taylor series
• N.B.

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• Runge-Kutta method
• Local error

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• Stochastic case

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• SRK Method

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Order and Convergence Results

• Definitions
• 1 Given rooted coloured tree with deterministic
  nodes and stochastic nodes,
• 2 Strong local order p if
• 3 Strong local order p if
• 4 Mean local order p if
• 5 Mean local order p if

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• Theorem
• A method has strong global order p if it has
  strong local order p and mean local order p.
• Remark For the Euler method,
• but it does not converge with order 1/2 for
  Stratonovich problems, only for Ito.
• Remark For p integer, d1, with coefficients
  dependent on , strong global order
  strong local order.
• Remark This does not hold for E1 which has
  strong local order 1.5, but not strong global
  order 1.5.

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Numerical Results

• G5 5-stage explicit SRK has strong global order
  1.5
• Test problem (Kloeden and Platen)
• Solution is
• Parameters used

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• 500 simulations were carried out.
• First problem is purely stochastic

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• Second problem is very stochastic requiring very
  small h.

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Conclusions

• Generic approach for constructing arbitrarily
  high order RK methods
• Extended recently to
• Predictor-corrector methods (Burrage and Tian -
  under preparation)
• Linear multistep methods (Brugnano and Burrages
  (2000))
• High order implicit methods (Tian and Burrage
  (2000))

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References

• L. Brugnano, K. Burrage and P. Burrage (2000)
  Adams type methods for the numerical solution of
  SODEs, to appear in BIT.
• K. Burrage and P.M. Burrage (1999) Order
  conditions of stochastic Runge-Kutta methods by
  B-series, submitted
• P.E. Kloeden and E. Platen (1992) The Numerical
  Solution of Stochastic Differential Equations,
  Springer-Verlag.
• T.Tian and K. Burrage (2000) High order implicit
  SRK methods for stiff SDEs, submitted.