Hello,

I am trying to integrate the transition density of the Brownian Motion - but Mathematica gives me values above one. If I understand correctly, the values that I get should all be less or equal to one.

What am I doing wrong?

bm[\[Mu]_, \[Sigma]_] := 
 ItoProcess[\[DifferentialD]x[
     t] == \[Mu] \[DifferentialD]t + \[Sigma] \[DifferentialD]w[t], 
  x[t], {x, 0}, t, w \[Distributed] WienerProcess[]]

In[46]:= pdf = PDF[gProc2[\[Mu], \[Sigma]][t], x]

Out[46]= E^(-((x - t \[Mu])^2/(2 t \[Sigma]^2)))/(Sqrt[2 \[Pi]] Sqrt[
 t \[Sigma]^2])

In[47]:= NIntegrate[E^(-((x - t \[Mu])^2/(2 t \[Sigma]^2)))/(
  Sqrt[2 \[Pi]] Sqrt[t \[Sigma]^2]), {t, 0, 5}, {x, 0, 
   5}] /. {\[Mu] -> 0.04, \[Sigma] -> 0.7}

During evaluation of In[47]:= NIntegrate::inumr: The integrand E^(-((x-<<1>>)^2/(2 t \[Sigma]^2)))/(Sqrt[2 \[Pi]] Sqrt[t \[Sigma]^2]) has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,5},{0,5}}. >>

Out[47]= 2.66889

Thank you in advance,

Edvin