Uniqueness Theorem of Solutions for Stochastic Differential Equation in the Plane  

作　　者：Liang Zongxia, Department of Applied Mathematics, Tsinghua University Beijing 100084, China 

出　　处：《Acta Mathematica Sinica,English Series》1998年第4期495-500,201+502-506,共12页数学学报（英文版）

基　　金：Supported by the National Science Foundation;the Postdoctoral Science Foundation of China

摘　　要：Let M = {M_z, z∈R₊²} be a continuous square integrable martingale and A = {A_z, z∈ R₊²} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX_z=α（z, X_z）dM₂+β（z,X_z）dA_z, z∈R₊², X_z=Z_z, z∈R₊², where R₊²=[0,+∞)×[0,+∞) and R₊² is its boundary, Z is a continuous stochastic process on R₊². We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.Let M = {M_z, z∈R₊²} be a continuous square integrable martingale and A = {A_z, z∈ R₊²} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX_z=α（z, X_z）dM₂+β（z,X_z）dA_z, z∈R₊², X_z=Z_z, z∈R₊², where R₊²=[0,+∞)×[0,+∞) and R₊² is its boundary, Z is a continuous stochastic process on R₊². We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.

关 键 词：Two-parameter S. D. E.  Two-parameter martingale ITO’s formula Pathwise uniqueness Gronwall’s-Bellman lemma 

分 类 号：O1[理学—数学]