ON THE RELATION BETWEEN ORDINARY AND STOCHASTIC DIFFERENTIAL EQUATIONS
ON THE RELATION BETWEEN ORDINARY AND STOCHASTIC DIFFERENTIAL EQUATIONS
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DOI：
10.1016/0020-7225(65)90045-5
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发表时间：
1965-07
影响因子：
6.6
通讯作者：
E. Wong;M. Zakai
中科院分区：
文献类型：
--
作者：
E. Wong;M. Zakai
本文考虑如下问题：设xt是随机微分方程：dx t= m [xt，t] dt+ σ [xt，t] dy t的解，其中yt是布朗运动过程.设xt（n）是随机微分方程的解，它是由随机微分方程用yt（n）代替yt得到的，其中yt（n）是布朗运动的连续分段线性逼近，yt（n）在n→∞时收敛于yt.如果x t是随机微分方程的解（在伊藤意义下），那么解的序列x t（n）是否收敛于x t？结果表明，答案一般是否定的。证明了xt（n）平均收敛于另一个随机微分方程的解dxt = m [xt，t] dt+ 1 2 σ [xt，t]（n σ [xt，t]/n xt）dt+ σ [xt，t] dy t.
The following problem is considered in this paper: Let x t be a solution to the stochastic differential equation: dx t= m [x t, t] dt+ σ [x t, t] dy t where y t is the Brownian motion process. Let x t (n) be the solution to the ordinary differential equation which is obtained from the stochastic differential equation by replacing y t with y t (n) where y t (n) is a continuous piecewise linear approximation to the Brownian motion and y t (n) converges to y t as n→∞. If x t is the solution to the stochastic differential equation (in the sense of Ito) does the sequence of the solutions x t (n) converge to x t? It is shown that the answer is in general negative. It is however, shown that x t (n) converges in the mean to the solution of another stochastic differential equation which is: dx t= m [x t, t] dt+ 1 2 σ [x t, t](∂ σ [x t, t]/∂ x t) dt+ σ [x t, t] dy t.