This is a classic problem in American option. Some discussion can be found in Shreve's book Ch 8 (8.4.1)

The problem can be summarized as find an exercise boundary, i.e. For example for a put option, we need to find a exercise boundary L(T-t), whenever the stock price S(t) falls bellow L(T-t), we should exercise. For your problem, it is the probability that lognormal distribution goes below L(T-t).

This problem is a little hard because it needs simultaneously solve PDE with several boundary conditions. So the common practice is to use finite difference (I know you don't like :-( ) to numerically find the boundary. Than you can calculate the probability.

More details, see Shreve's book. I think this is a popular issue. There should be many research papers talking about how to get the exercise boundary more efficiently (perhaps some analytical solutions :-) ).

best,

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问题不太专业，影响因素太多了，比如假设的标的资产的变化形式，是否是完全市场等等，你得具体问题具体分析

已经改正了，请不吝赐教

在二叉树的框架下可以得到近似的最优执行边界，并且可以知道那些路径穿过了边界，执行概率也就算出来了。不过这种方法不够“高大上”。
在连续情境下，执行概率问题可以抽象为一个“首达时”问题，不过时变的边界条件比教科书上的反射原理复杂多了。可以推荐一些关于执行概率的论文吗？
执行概率对定价没太大帮助，但是对交易来说很有参考价值，特别是那些期望卖出裸期权的投机者。
把执行概率和期望收益结合起来编制一个效用函数