In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S . In the classical Black–Sholes model for a stock price, Theorem 4.3 of [S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 1991, 2, 1–14] states that if the family of integral equations (parametrized by the variable S ) holds for all S≤b⁢(t){S\leq b(t)} with a candidate function b⁢(t){b(t)}, then this b⁢(t){b(t)} must coincide with the American put early exercise boundary c⁢(t){c(t)}. We generalize Peskir’s result [G. Peskir, On the American option problem, Math. Finance 15 2005, 1, 169–181] to state that if the candidate function b⁢(t){b(t)} satisfies one particular integral equation (which corresponds to the upper limit S=b⁢(t){S=b(t)}), then all other integral equations (corresponding to S , S≤b⁢(t){S\leq b(t)}) will be automatically satisfied by the same function b⁢(t){b(t)}.

中文翻译：

---

关于早期运动边界的非线性Volterra积分方程的一个注记

在本文中，我们研究一维扩散模型中具有恒定利率和恒定股利收益且局部波动取决于股票价格的一维扩散模型中美国认沽期权的早期行使边界所满足的非线性Volterra积分方程。时间t和当前股票价格S。在经典的Black-Sholes股票价格模型中，[SD Jacka，最优止损和美国看跌期权数学]定理4.3。Finance 1 1991，2，1–14]指出，如果对所有带有候选函数b S的S≤b⁢（t）{S \ leq b（t）}成立积分方程组（由变量S表示） （t）{b（t）}，则此b⁢（t）{b（t）}必须与美国投放的早操边界c⁢（t）{c（t）}一致。我们概括了佩斯基尔的结果[G. Peskir，关于美式期权问题，数学。财经15 2005，1，