A COMPARISON THEOREM FOR SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS
A COMPARISON THEOREM FOR SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS
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DOI：
10.1090/s0002-9939-1984-0746100-9
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发表时间：
1984-04
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影响因子：
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通讯作者：
Z. Huang
中科院分区：
文献类型：
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作者：
Z. Huang
利用偏微分方程的广义样本解，建立了一个偏微分方程与两个确定性偏微分方程的比较定理。利用这个定理，我们可以比较两个具有不同扩散系数的随机微分方程的解，并得到扩散过程路径的一些渐近估计。在研究随机微分方程解的过程中，一致性定理与确定性方程一样，是非常有力的工具。但是到目前为止，这些定理中的大多数都是处理那些具有相同扩散系数的随机微分方程（参见。（1，3-7，9，10）），除了在（8）中讨论了涉及两个不同扩散系数的非常特殊的情况（参见实施例2）。在本文中，我们使用的方法，广义样本解的偏微分方程（cf。（5，11））建立了一种新的比较定理，其中一个微分方程与两个确定性常微分方程进行比较.这里的条件比（5和6）中的条件弱，证明也简单得多。另一方面，允许不连续的右侧。因此，它似乎更适合于随机最优控制问题。我们从一个引理开始开始。莱姆设函数f（t，x）和f（t，x）定义在R ~ 2中的某个区域G上，满足Caratheodory条件，即它们在t上可测，在x上连续，且被区域G中的局部可积函数m（t）支配.设（in，xo）an（^（*o，？o）是G中的两个点，使得xq
A new kind of comparison theorem in which an SDE is compared with two deterministic ODEs is established by means of the generalized sample solutions of SDEs. Using this theorem, we can compare solutions of two SDEs with different diffusion coefficients and obtain some asymptotic estimations for the paths of diffusion processes. In the investigation of solutions of stochastic differential equations (SDEs), com- parison theorems are very powerful tools as in the case for deterministic ones. But so far most of these theorems have dealt with those SDEs with the same diffusion coefficient (cf. (1, 3—7, 9, 10)) except in (8) where a very special case involving two different diffusion coefficients has been discussed (cf. Example 2). In this paper, we use the method of generalized sample solutions of SDEs (cf. (5, 11)) to establish a new kind of comparison theorem in which an SDE is compared with two deterministic ODEs. The conditions imposed here are weaker than those in (5 and 6) and the proof is much simpler. On the other hand, a discontinuous right-hand side is allowed. So it seems more appropriate to the stochastic optimal control problems. We begin with a lemma. LEMMA. Assume that two functions f{t,x) and f(t,x) are defined on some domain G in R2 satisfying the Caratheodory conditions, that is, they are measurable in t, continuous in x and dominated by a locally integrable function m(t) in the domain G. Let (in,xo) an(^ (*o,?o) be two points in G such that xq < xq, x{t) be any solution to the initial value problem