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Using the Dynkin's formula, prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$

where $F(s,t)=2\int_{x-x_0}^{\infty}\frac{1}{\sqrt{2\pi t}}e^{-\frac{u^2}{2t}}du-1$

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  • $\begingroup$ This can be done by simple calculus, why do you need Dynkin's formula? $\endgroup$ – Gordon May 23 '18 at 16:55
  • $\begingroup$ My idea is to use the Dynkin formula to then deduce the distribution of the maximum process of Brownian motions $\endgroup$ – FunnyBuzer May 23 '18 at 17:02
  • $\begingroup$ Assume that the above holds, how to use this result to obtain the distribution of the maximum process of Brownian motions? $\endgroup$ – FunnyBuzer May 24 '18 at 13:05
  • $\begingroup$ I do not know this. You may have a look of the this book. $\endgroup$ – Gordon May 24 '18 at 13:26

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