# Prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$

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

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