Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</...Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</sup>} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX<sub>z</sub>=α（z, X<sub>z</sub>）dM<sub>2</sub>+β（z,X<sub>z</sub>）dA<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, X<sub>z</sub>=Z<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, where R<sub>+</sub><sup>2</sup>=[0,+∞)×[0,+∞) and R<sub>+</sub><sup>2</sup> is its boundary, Z is a continuous stochastic process on R<sub>+</sub><sup>2</sup>. 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.展开更多
We prove that two parameter smooth continuous martingales have ∞-modification and establish a Doob’s inequality in terms of(p,r)-capacity for two parameter smooth martingales.
基金Supported by the National Science Foundationthe Postdoctoral Science Foundation of China
文摘Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</sup>} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX<sub>z</sub>=α（z, X<sub>z</sub>）dM<sub>2</sub>+β（z,X<sub>z</sub>）dA<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, X<sub>z</sub>=Z<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, where R<sub>+</sub><sup>2</sup>=[0,+∞)×[0,+∞) and R<sub>+</sub><sup>2</sup> is its boundary, Z is a continuous stochastic process on R<sub>+</sub><sup>2</sup>. 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.
文摘We prove that two parameter smooth continuous martingales have ∞-modification and establish a Doob’s inequality in terms of(p,r)-capacity for two parameter smooth martingales.
