Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients. (English) Zbl 1102.60059
The paper investigates correct ways of discretization for stochastic differential equations (SDEs) with coefficients that are not bounded, and are only locally but not globally Lipschitz, on a fixed finite time horizon. A class of SDEs is considered with a Lyapunov function, which guarantees non-explosion. It is proposed to change the coefficients of the SDE outside a large ball of radius \(R\), so that the new ones satisfy a global Lipschitz condition. Naturally, with a large probability, solution of a new equation coincides with solution of the original one, and a small complementary probability \(\varepsilon\) possesses an explicit bound via the corresponding Lyapunov function. It is proved that any weakly convergent method on globally Lipschitz coefficients, combined with this proposed change outside a large ball, possesses a bound \(Ch^p + \varepsilon\), where \(h\) is a discretization step, and \(p\) is the order of the methods on globally Lipschitz equations. Examples and numerical simulation results are provided.
Reviewer: Alexander Yu. Veretennikov (Leeds)
MSC:
60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
65C35 | Stochastic particle methods |
60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |