由标值点过程驱动的倒向随机微分方程：一种基本方法及其在最优控制中的应用
Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control
摘要：
研究了一类有界区间上的倒向随机微分方程，其中驱动噪声是一个显著的或多变量的点过程。假设跳跃时间是完全不可达的，并且技术条件成立（见下面的假设（A）），我们证明了系数在利普希茨条件下的存在性和唯一性结果。一些反例表明，我们的假设确实是必要的。我们使用一种新的方法，允许减少到（有限或无限）系统的确定性微分方程，从而避免使用鞅表示定理，并允许潜在的使用标准的数值方法。最后，我们应用主要结果来解决一个标值点过程的最优控制问题，制定在一个经典的方式。
英文摘要：
We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter-examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.
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