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Can anyone help me solve this problem. I have no idea where to even start on it. Link inside stock option problem

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    Is $F$ an arbitrary distribution?2011-02-28

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You are asked to prove that $ \int_{ - \infty }^\infty {V_{n - 1} (s + x)dF(x)} > s - c, $ for all $n \geq 1$. For $n=1$, substituting from the definition of $V_0$, you need to show that $ \int_{ - \infty }^\infty {\max \lbrace s + x - c,0\rbrace dF(x)} > s - c. $ For this purpose, first note that $ \max \lbrace s + x - c,0\rbrace \ge s + x - c. $ Then, by linearity of the integral, you can consider the sum $ \int_{ - \infty }^\infty {(s - c)dF(x)} + \int_{ - \infty }^\infty {xdF(x)} , $ from which the assertion for $n=1$ follows.

To complete the inductive proof, substituting from the definition of $V_n$, you need to show that $ \int_{ - \infty }^\infty {\max \bigg\lbrace s + x - c,\int_{ - \infty }^\infty {V_{n - 1} (s + x + u)dF(u)} \bigg\rbrace dF(x)} > s - c, $ under the induction hypothesis that, for any $s \in \mathbb{R}$, $ \int_{ - \infty }^\infty {V_{n - 1} (s + x)dF(x)} > s - c. $ For this purpose, recall the end of the proof for $n=1$.