Abstract
Solving stochastic differential equations (SDEs) numerically, explicit Euler-Maruyama (EM) schemes are used most frequently under global Lipschitz conditions for both drift and diffusion coefficients. In contrast, without imposing the global Lipschitz conditions, implicit schemes are often used for SDEs but require additional computational effort; along another line, tamed EM schemes and truncated EM schemes have been developed recently. Taking advantages of being explicit and easily implementable, truncated EM schemes are proposed in this paper. Convergence of the numerical algorithms is studied, and pth moment boundedness is obtained. Furthermore, asymptotic properties of the numerical solutions such as the exponential stability in pth moment and stability in distribution are examined. Several examples are given to illustrate our findings.
Highlights
In this paper, we study numerical solutions of d-dimensional stochastic differential equations (SDEs) of the form dx(t) = f (x(t)) dt + g(x(t)) dB(t), t ≥ 0, x(0) = x0, (1.1)where B(t) is an m-dimensional Brownian motion and f : Rd → Rd, g : Rd → Rd×m, which satisfy a local Lipschitz condition, namely, for any N > 0 there is a constant CN such that f (x) − f (y) ∨ g(x) − g(y) ≤ CN x − y (1.2)for any x, y ∈ Rd with |x|∨|y|≤ N
In addition to obtaining the asymptotic pth moment convergence and moment boundedness we consider the approximations to the invariant distributions in infinite horizon
The second moment of the classical EM numerical solution diverges to infinity in an infinite time interval for any given step size and an initial value dependent on the step size. As their counterparts of analytic solutions, we show that our explicit schemes will preserve the asymptotic moment boundedness as well as asymptotic stability for a large class of nonlinear SDEs including (1.3) and (1.4) under Assumptions 5.1, 6.1, 7.1
Summary
Based on the motivation above, we construct implementable explicit EM schemes for SDEs with only local Lipschitz drift and diffusion coefficients and establish their convergence. We answer the question of Higham et al positively by requiring only that the drift and diffusion coefficients are locally Lipschitz and satisfy a structure condition (Assumption 2.1) for the pth moment boundedness of the exact solution for some p ∈ (0, +∞). In this paper, adopting the truncation idea from the study by Mao (2015) and using a novel approximation technique, we construct several explicit schemes under certain assumptions on the coefficients of the SDEs and derive convergence results in both finite and infinite time intervals. The pth moment of our explicit numerical solution is bounded for the SDEs with only local Lipschitz drift and diffusion coefficients.
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