非Lipschitz条件下由一般鞅驱动的倒向随机微分方程解的存在性

摘要

经典的倒向随机微分方程以布朗运动做为干扰源，布朗运动是一种理想化的随机模型，从而使倒向随机微分方程的应用受到了限制。文中研究了以连续局部鞅为干扰源的倒向随机微分方程，在生成元满足一种非Lipschitz条件下，通过构造一个函数列的方法，利用Lebesgue's控制收敛定理和常微分方程的比较定理，证明了其解是存在的并且是唯一的，对经典的倒向随机微分方程进行了推广。%　　The classical backward stochastic differential equations (BSDE) theory is taken the Brownian motion as the noise source, but the Brown motion is one kind of extreme idealized model, which causes limitations of the BSDE theory in application. This paper studies the backward stochastic differential equation which is taken continuous local martingale as the noise source. The authors of the paper conclude a general existence and uniqueness of the solutions under non-Lipschitz condition on the generator by adopting the construction of function sequence, Lebesgue's dominated convergence theorem and the comparison of ordinary differential equation. The authors have also conducted a substantial extension of the classical backward stochastic differential equations.