| Abstract: |
| We propose a new class of symplectic schemes for the strong approximation of the solutions of stochastic Hamiltonian systems. The proposed schemes are a trade-off between the stochastic Runge-Kutta methods and the symplectic methods based on generating functions. Symplectic stochastic Runge-Kutta methods use only first order derivatives of the Hamiltonians, but they satisfy many algebraic conditions, which makes strong orders above 1 hard to obtain for general stochastic Hamiltonian systems with multiplicative noise. Symplectic schemes of strong order 1.5 or higher can be obtained using generating functions, but they require higher order derivatives of the Hamiltonians. Here we propose a new family of symplectic schemes constructed by defining a generating function similar with the ones associated to symplectic stochastic Runge-Kutta methods. The proposed schemes are a generalization of the implicit midpoint rule, are symplectic by construction, and use derivatives of at most second order. We derive schemes of strong order 1.5 for general stochastic Hamiltonian systems and study their long time behaviour through numerical experiments. |
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