Abstract. We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients. Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. We prove that accuracy of any method of weak order p is estimated by ε + O(h p ), where ε can be made arbitrarily small with increasing radius of the sphere. The results obtained are supported by numerical experiments. 1. Introduction. Stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients possessing unique solutions make up a very important class in applications. For instance, Langevin-type equations and gradient systems with noise belong to this class [10,9,1,5,11]. At the same time, most numerical methods for SDEs are derived under the global Lipschitz condition [3,7]. If this condition is violated, the behavior of many standard numerical methods in the whole space can lead to incorrect conclusions (see, for instance, [9,1,5,11,4]). This situation is very alarming since we are forced to refuse many effective methods and/or to resort to some comparatively complicated and inefficient numerical procedures. In [6] (see also Example 3.3 here), applying an explicit quasi-symplectic method of weak approximation to a Langevin equation with nonglobally Lipschitz coefficients for calculating an ergodic limit, the authors found an explosive behavior of some approximate trajectories. The explosions are observed outside of a comparatively large sphere after a relatively large time and very rarely. Clearly, the exploding approximate trajectories badly reproduce the actual behavior of the considered system. We have also found that if these rare trajectories are discarded, then the explicit quasi-symplectic method gives much better results than the implicit Euler method, which does not have any exploding trajectories. From the heuristic point of view, this is rather natural. Roughly speaking, the value of an ergodic limit depends, on the whole, on the behavior of trajectories in a bounded (though large) domain on a finite (though large) time interval. Consequently, any method that is effective for systems with globally Lipschitz coefficients has to work well for systems with nonglobally Lipschitz coefficients as well if one rejects a small number of "bad" trajectories.In this paper, we propose a new concept which allows us to apply any method of weak approximation to a very broad class of SDEs with nonglobally Lipschitz coef-

Abstract. Stochastic Hamiltonian systems with multiplicative noise, phase flows of which preserve symplectic structure, are considered. To construct symplectic methods for such systems, sufficiently general fully implicit schemes, i.e., schemes with implicitness both in deterministic and stochastic terms, are needed. A new class of fully implicit methods for stochastic systems is proposed. Increments of Wiener processes in these fully implicit schemes are substituted by some truncated random variables. A number of symplectic integrators is constructed. Special attention is paid to systems with separable Hamiltonians. Some results of numerical experiments are presented. They demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones.

Abstract. Hamiltonian systems with additive noise possess the property of preserving symplectic structure. Numerical methods with the same property are constructed for such systems. Special attention is paid to systems with separable Hamiltonians and to second-order differential equations with additive noise. Some numerical tests are presented. Key words.Hamiltonian systems with additive noise, symplectic integration, mean-square methods for stochastic differential equations

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A number of new layer methods for solving semilinear parabolic equations and reaction-diffusion systems is derived by using probabilistic representations of their solutions. These methods exploit the ideas of weak sense numerical integration of stochastic differential equations. In spite of the probabilistic nature these methods are nevertheless deterministic. A convergence theorem is proved. Some numerical tests are presented.

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We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients. Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. We prove that accuracy of any method of weak order p is estimated by ε + O(h p), where ε can be made arbitrarily small with increasing radius of the sphere. The results obtained are supported by numerical experiments. 1. Introduction. Stochastic differential equations (SDEs) with nonglobally Lip-schitz coefficients possessing unique solutions make up a very important class in applications. For instance, Langevin-type equations and gradient systems with noise belong to this class [10, 9, 1, 5, 11]. At the same time, most numerical methods for SDEs are derived under the global Lipschitz condition [3, 7]. If this condition is violated, the behavior of many standard numerical methods in the whole space can lead to incorrect conclusions (see, for instance, [9, 1, 5, 11, 4]). This situation is very alarming since we are forced to refuse many effective methods and/or to resort to some comparatively complicated and inefficient numerical procedures. In [6] (see also Example 3.3 here), applying an explicit quasi-symplectic method of weak approximation to a Langevin equation with nonglobally Lipschitz coefficients for calculating an ergodic limit, the authors found an explosive behavior of some approximate trajec-tories. The explosions are observed outside of a comparatively large sphere after a relatively large time and very rarely. Clearly, the exploding approximate trajectories badly reproduce the actual behavior of the considered system. We have also found that if these rare trajectories are discarded, then the explicit quasi-symplectic method gives much better results than the implicit Euler method, which does not have any exploding trajectories. From the heuristic point of view, this is rather natural. Roughly speaking, the value of an ergodic limit depends, on the whole, on the behavior of trajectories in a bounded (though large) domain on a finite (though large) time interval. Consequently, any method that is effective for systems with globally Lipschitz coefficients has to work well for systems with nonglobally Lipschitz coefficients as well if one rejects a small number of "bad" trajectories. In this paper, we propose a new concept which allows us to apply any method of weak approximation to a very broad class of SDEs with nonglobally Lipschitz coef