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Shreve 7.2.2
Under Probability P, W(T) is a standard Brownian Motion
while X(T)=W(T)+aT is a Brownian Motion plus drift.
M(T)=Max{X(s); 0<s<T}
\(P \{M(T)\leq m\} = N(\frac{m-aT}{\sqrt{T}})-e^{2am}N(\frac{-m-aT}{\sqrt{T}}),m>0\)
Can we differentiate the above to get first passage time density?
\(\frac{\partial}{\partial T}P\{M(T)\leq m\}\)
I corrected X(T) by M(T)
Under Probability P, W(T) is a standard Brownian Motion
while X(T)=W(T)+aT is a Brownian Motion plus drift.
M(T)=Max{X(s); 0<s<T}
\(P \{M(T)\leq m\} = N(\frac{m-aT}{\sqrt{T}})-e^{2am}N(\frac{-m-aT}{\sqrt{T}}),m>0\)
Can we differentiate the above to get first passage time density?
\(\frac{\partial}{\partial T}P\{M(T)\leq m\}\)
I corrected X(T) by M(T)