Numerical Integration of Stochastic Differential Equations

Abstract

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-

Convergence Rates for Adaptive Weak Approximation of Stochastic Differential Equations

“…On the other hand, letting ≡ t ∈ L 2 dt×P 0 T × t is stochastic, positive and piecewise constant on t k leads to r n = constant for all time steps n and for all realization (30) which sets the basis for the refinement procedure with stochastic time steps. Thus, the adaptive algorithm with stochastic time steps uses the optimal conditions (27) and (30) to construct the mesh, which may be different for each realization. On the other hand, the adaptive algorithm with deterministic time steps uses the optimal conditions (27) and (29) to construct the mesh, which is the same for all realizations.…”

“…Thus, the adaptive algorithm with stochastic time steps uses the optimal conditions (27) and (30) to construct the mesh, which may be different for each realization. On the other hand, the adaptive algorithm with deterministic time steps uses the optimal conditions (27) and (29) to construct the mesh, which is the same for all realizations. Note that (29)- (30) do not take the sign of the error density into account and, in this sense, our time steps are optimal only for error densities of one sign.…”

“…The optimal conditions (27), (30) and the restriction (15) motivate that the goal of the adaptive algorithm, with stochastic time steps, for each realization, is to construct a time partition t of 0 T such that…”

“…The Monte Carlo Euler method approximates the unknown process X by the Euler approximation X t n (cf. Kloeden and Platen [25] and Milstein [27]), which is a time discretization based on the nodes 0 = t 1 < t 2 < · · · < t N +1 = T where X t n+1 − X t n = t n a t n t n X t n + 0 =1 W n b t n X t n (2) with time increments t n ≡ t n+1 − t n and Wiener increments W n ≡ W t n+1 − W t n for n = 1 2 N and = 1 2 0 . The aim of the adaptive algorithm is to choose the size of the time steps, t n ,…”

Numerical Simulation of the Heston Model under Stochastic Correlation

“…The discretization of (34) by applying the Milstein scheme Milstein (1974) will be the same to (35), since all the derivatives included in the coefficients of the double integral terms (with respect to BMs) by the Milstein scheme are equal to zero. Moreover, we remark thatŜ t = exp(x t ) with the discretized processx t in (35) is a martingale, and any types of stochastic correlation processes can be straightforwardly employed within this scheme.…”