THE BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL-INEQUALITIES
THE BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL-INEQUALITIES
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DOI：
10.1007/bf01246336
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发表时间：
1995-12-01
影响因子：
2
通讯作者：
MANTHEY, R
中科院分区：
文献类型：
--
作者：
ASSING, S;MANTHEY, R
设 X 和 Z 分别为小于或等于 dZ(t) 的随机微分不等式 dX(t) 小于或等于 a(t,X(t))dt + sigma(t,X(t))dW(t) 和 b(t,Z(t))dt + sigma(t,Z(t))dW(t) 的 R(d) 值解，且 R(m) 值固定维纳过程 W。在本文中，我们给出了 a、b 和 sigma 的条件，在该条件下，关系 X(0) 小于或等于初始值的 Z(0) 导致解之间的关系以概率为 1。我们进一步讨论我们的条件总体上是否可以被削弱。然后我们处理诸如随机微分不等式的“最大/最小解”之类的概念。利用比较结果，我们得出了此类“解”存在的充分条件以及一些 Gronwall 型估计。
Let X and Z be R(d)-valued solutions of the stochastic differential inequalities dX(t) less than or equal to a(t,X(t))dt + sigma(t,X(t))dW(t) and b(t,Z(t))dt + sigma(t,Z(t))dW(t) less than or equal to dZ(t), respectively, with a fixed R(m)-valued Wiener process W. In this paper we give conditions on a,b and sigma under which the relation X(0) less than or equal to Z(0) of the initial values leads to the same relation between the solutions with probability one. Further we discuss whether in general our conditions can be weakened or not. Then we deal with notions like 'maximal/minimal solution' of a stochastic differential inequality. Using the comparison result we derive a sufficient condition for the existence of such 'solutions' as well as some Gronwall-type estimates.