A Note on the Carathéodory Approximation Scheme for Stochastic Differential Equations under G-Brownian Motion
نویسنده
چکیده
The Carathéodory approximation scheme was introduced by the Greek mathematician named Constantine Carathéodory in the early part of 20th century for ordinary differential equations (Chapter 2 of [1]). Later this was extended by Bell and Mohammad to stochastic differential equations [2] and then by Mao [3, 4]. Generally, the solutions of stochastic differential equations (SDEs) do not have explicit expressions except the linear SDEs. We therefore look for the approximate solutions instead of the exact ones such as the Picard iterative approximate solutions etc. Practically, to compute Xk(t) by the Picard approximation, one need to compute X0(t),X1(t), . . . ,Xk−1(t), which involve a lot of calculations on Itô’s integrals. However, by the Carathéodory approximation we directly compute Xk(t) and do not need the above mentioned stepwise iterations, which is an admirable advantage as compared to the Picard approximation [5]. The theory of G-Brownian motion and the related Itô’s calculus was introduced by Peng [6]. He developed the existence and uniqueness of solutions for stochastic differential equations under G-Brownian motion (G-SDEs) under the Lipschitz conditions via the contraction method [6, 7]. While by the Picard approximation the existence theory for G-SDEs was established by Gao [8] and then by Faizullah and Piao with the method of upper and lower solutions [9]. Also see [10]. In this paper, the Carathéodory approximation scheme for G-SDEs is entrenched. It is shown that under some suitable conditions the Carathéodory approximate solutions Xk(t), k ≥ 1, converge to the unique solution X(t) of the G-SDEs in the sense that
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تاریخ انتشار 2012