AI 中文总结
研究有界状态空间中齐次扩散鞅，建立其与伯努利 - 杜布鞅的等价关系，证明伯努利极限渐近性，阐明相关边界分类等关系，通过\(\Phi\)鞅等进行理论说明及信用风险建模应用。
AI 中文摘要
我们研究在有界状态空间\(D = [a,b]\)中演化的齐次扩散鞅，其中\(a\)和\(b\)是扩散系数的零点。我们称形如\(Z_t=\mathbb{E}[B\mid\mathcal{F}_t]\)（\(B\)为伯努利随机变量）的过程为伯努利 - 杜布鞅。主要结果建立了完全等价性：每个这样的扩散鞅都是伯努利 - 杜布鞅（定理2），反之，布朗滤波上的每个连续时间齐次马尔可夫伯努利 - 杜布鞅都源于这样的扩散（定理3）。直观原因是有界鞅在积累方差时期望恒定，所以收敛到具有给定均值和范围的最大方差分布，即伯努利分布。我们进一步表明这个伯努利极限是真正渐近的：对于任何固定的有限时间范围\(T\)，尚未到达边界的概率严格为正（定理4），即使单个边界是可达的。我们阐明了费勒边界分类、路径wise SDE框架和鞅约束之间的关系，表明鞅性质迫使在任何可达边界处吸收。该理论通过\(\Phi\)鞅、雅可比鞅以及信用风险建模应用进行说明。
英文摘要
We study homogeneous diffusion martingales evolving in a bounded state space $D=[a,b]$, where $a$ and $b$ are zeros of the diffusion coefficient. We call a process of the form $Z_t=\mathbb{E}[B\mid\mathcal{F}_t]$, with $B$ a Bernoulli random variable, a Bernoulli-Doob martingale. Our main results establish a complete equivalence: every such diffusion martingale is a Bernoulli-Doob martingale (Theorem 2) and, conversely, every continuous time-homogeneous Markov Bernoulli-Doob martingale on a Brownian filtration arises from such a diffusion (Theorem 3). The intuitive reason is that a bounded martingale has constant expectation while accumulating variance, so it converges to the maximum-variance distribution with given mean and range, namely the Bernoulli. We further show that this Bernoulli limit is truly asymptotic: for any fixed finite horizon $T$, the probability of not yet having reached the boundary is strictly positive (Theorem 4), even when the individual boundaries are accessible. We clarify the relationship between Feller's boundary classification, the pathwise SDE framework, and the martingale constraint, showing that the martingale property forces absorption at any attainable boundary. The theory is illustrated with the $Φ$-martingale, the Jacobi martingale, and applications to credit-risk modelling.
