Measure Solution Of BSDE With Time Delayed Generator

We study the measure solution of stochastic ordinary differential equation with time delayed generator. First, for better understanding, we improve and complete the proof of existence and uniqueness theorem of adapted solution in existing literature. This theorem holds under the assumptions that time horizon and Lipschitz constant can be arbitrarily small. Then, we illustrate through citing two examples that traditional comparison theorem and measure solution can not exist under general conditions. By adding new requirements, we show a new comparison theorem and two new proofs of measure solution. In the latter part of this paper, we list some known facts about this equation, including for linear time delayed generator, existence and uniqueness may also hold for an arbitrary time horizon and for arbitrary Lipschitz constant, the boundness between solution and terminal condition, and the BMO martingale property.