AI 中文总结
该研究针对常数波动率的非线性均值回复一维扩散过程，推导了Novikov准则中积分平方漂移项指数矩的精确阈值，明确了不同回复率下矩的有限性条件及临界边界。
AI 中文摘要
我们研究了Novikov准则中出现的积分平方漂移项的指数矩，针对具有常数波动率σ、漂移项λ严格均值回复但尾部可能快速变陡的一维扩散过程。通过局部测度变换将问题简化为单一势函数，其中漂移项仅通过其导数即局部回复率q=-λ'进入。由此得到两个结论：第一，在一类自然的陡回复漂移项上，Novikov系数a=1/2是唯一使得时间跨度起作用的系数：低于该系数时，所有时间跨度上的矩均有限；高于该系数时，所有时间跨度上的矩均无限。第二，在该临界线上，回复率的增长情况起决定作用：亚二次回复率在所有时间跨度上给出有限矩，主导所有二次项的回复率在所有正时间跨度上给出发散矩；当回复率恰好以二次方式增长且极限曲率为κ∞时，我们得到精确边界κ∞σ²T²=π²：低于该边界时矩有限，高于时无限，边界处对于三次和仿射三次漂移项为无限；该常数是布朗桥的第一狄利克雷特征值，其出现是因为四次终端势约束了变换后的路径。三次回复是典型情况，临界时间跨度为κ∞σ²T²=π²/3；指数陡回复严格位于临界尺度之上。此外，消除漂移项的随机指数仍为真正的鞅，因此经典矩准则的临界性与其旨在验证的指数的临界性不同。
英文摘要
We determine the finiteness boundary of the Novikov exponential moment for a one-dimensional constant-volatility mean-reverting diffusion. A localised change of measure cancels the squared drift and leaves a Brownian Feynman-Kac functional with potential $q/2$, where $q=-λ'$. If $q(θ+y)\simκ_\infty y^2$, the exact spectral boundary is $κ_\inftyσ^2T^2=π^2$; for cubic drift it becomes $cσ^2T^2=π^2/3$, and equality is divergent. On the spectral equality surface, the first lower-order transition occurs at power $4/3$, where an explicit coefficient separates the two sides. For symmetric finite pure-power tails, exact tuning generates the recursion $β_n=1+3^{-(n+1)}$. We prove that this recursion gives a complete classification of the class. Each individual tail is decided after finitely many comparisons, but the required depth is unbounded: arbitrarily long common critical prefixes can lead to opposite outcomes. The drift-removing stochastic exponential nevertheless remains a true martingale; under the physical law it has no higher moments on the steep-drift class, while the reverse density is essentially bounded.
Comments48 pages, 3 figures. Ancillary numerical Python script included
